KPZ equation limit of stochastic higher spin six vertex model
Yier Lin
TL;DR
This paper proves that the stochastic higher spin six vertex model converges to the KPZ equation under weak asymmetry for general spin parameters, extending previous results limited to the case I=J=1.
Contribution
It generalizes the convergence result of the SHS6V model to the KPZ equation from the specific case I=J=1 to arbitrary integer spins I, J.
Findings
Convergence of SHS6V to KPZ under weak asymmetry.
Extension of previous results to general spins I, J.
Validation of KPZ universality for broader class of models.
Abstract
We consider the stochastic higher spin six vertex (SHS6V) model introduced in [Corwin-Petrov, 2016] with general integer spin parameters . Starting from near stationary initial condition, we prove that the SHS6V model converges to the KPZ equation under weakly asymmetric scaling. This generalizes the result of [Corwin-Ghosal-Shen-Tsai, 2018] from to general .
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KPZ equation limit of stochastic higher spin six vertex model
Yier Lin
Abstract.
We consider the stochastic higher spin six vertex (SHS6V) model introduced by Corwin and Petrov [CP16] with general integer spin parameters . Starting from near stationary initial condition, we prove that the SHS6V model converges to the Kardar–Parisi–Zhang (KPZ) equation under weakly asymmetric scaling. This generalizes the result in [CGST18, Theorem 1.1] from to general .
Department of Mathematics, Columbia University, Email: [email protected]
Contents
- 1 Introduction
- 2 The bi-infinite model
- 3 Markov duality
- 4 Integral formula for the two particle transition probability
- 5 Microscopic Hopf-Cole transform and SHE
- 6 Tightness and proof of Theorem 5.6
- 7 Estimate of the two particle transition probability
- 8 Proof of Proposition 6.8 via self-averaging
- A Stationary distribution of the SHS6V model
- B KPZ scaling theory
1. Introduction
1.1. KPZ equation and weak KPZ universality
The KPZ equation is the following non-linear stochastic partial differential equation (SPDE) introduced in the seminal work [KPZ86], which describes the random evolution of an interface that has the property of relaxation and lateral growth
[TABLE]
Here is the space time white noise, which could be formally understood as a Gaussian field with covariance function \mathbb{E}\big{[}\xi(t,x)\xi(s,y)\big{]}=\delta(t-s)\delta(x-y), where is the Dirac delta function.
Care is needed to make sense of (1.1) due to the nonlinearity . The Hopf-Cole solution to the KPZ equation is defined by
[TABLE]
where is the mild solution of the Stochastic Heat Equation (SHE)
[TABLE]
So long as is (almost surely) positive, [Mue91] proved that remains positive for all and . This justifies the well-definedness of (1.2). Other equivalent definitions of the solution are given by regularity structure [Hai14], paracontrolled distribution [GP17] or the notion of energy solution [GJ14, GP18].
It is well-known that there is no non-trivial scaling under which the KPZ equation is invariant in law. More precisely, if we define , using the scaling of space-time white noise (in law), then
[TABLE]
It is clear that there is no such that the coefficients in the above equation match with those in (1.1). However, if we simultaneously scale some of the parameters , , , it is possible that the KPZ equation remains unchanged: such scaling is called weak scaling. It is thus natural to believe that the KPZ equation is the weak scaling limit of microscopic models with similar properties such as relaxation and lateral growth. Roughly speaking, this is the weak universality of the KPZ equation, see [Cor12, Qua11] for an extensive survey. We emphasize that the weak universality of the KPZ equation should be distinguished from KPZ universality, which says that without tuning of the parameter of the model, the microscopic system converges to a universal limit called KPZ fixed point under scaling, see [MQR16, DOV18, BL19] for some recent progress and breakthroughs in identifying the KPZ fixed point.
The weak universality of the KPZ equation has been verified for a number of interacting particle systems. The first result was given in the work of [BG97], for Asymmetric Simple Exclusion Process (ASEP). For more results of the weak universality of KPZ equation, see Section 1.5.3 of [CGST18] for a brief review.
Recently [CGST18, Theorem 1.1] proved that under weak asymmetric scaling (which corresponds to taking and in (1.3)), the stochastic six vertex model converges to the KPZ equation. In this paper, we consider stochastic higher spin six vertex model (SHS6V) model introduced in [CP16]111The SHS6V model has vertical and horizontal spin parameters . The stochastic six vertex model is a degeneration of it by taking .. We prove that under similar weak asymmetric scaling, the SHS6V model converges to the KPZ equation. This extends the result of [CGST18, Theorem 1.1] to the full generality. We like to emphasize that there are some significant new complications in our case compared with [CGST18], see Section 1.4 for discussion.
Before ending this section, we remark that there might be other SPDEs (besides the KPZ equation) arising from the vertex model. For instance, it was shown in [BG18, ST19] that under a different scaling, the stochastic six vertex model converges to the solution of the stochastic telegraph equation. It is interesting to ask whether the SHS6V model converges to other SPDEs, this question is left for future work.
1.2. The SHS6V model
The SHS6V model introduced in [CP16] (also see [Bor17]) belongs to the family of vertex models which themselves are examples of quantum integrable systems. In general, the -matrix (which can be thought of as the weights associated to the vertex) are not stochastic. [GS92, BCG16] studied the stochastic six vertex model, which is a stochastic version of the six vertex model introduced by [Pau35]. The authors of [CP16] worked with the -matrices, which is a stochastic version of the -matrices222See [CP16, Remark 2.2] for more discussion of the relation between -matrices and -matrices. and they defined the SHS6V model. The stochasticity allows us to define the vertex model on the entire line as an interacting particle system which follows sequential Markov update rule. Moreover, the -matrices in [CP16] satisfy the Yang-Baxter equation which implies the integrability of the model. In particular, the transfer matrices are diagonalizable by a complete set of Bethe ansatz eigenfunctions [BCPS15, CP16]. The model also enjoys Markov duality. The stochastic -matrices of the SHS6V model have four parameters, by specifying which the SHS6V model degenerates to known integrable systems such as stochastic six vertex model, ASEP, q-Hahn TASEP, q-TASEP. Indeed, it is on top of a hierarchy of KPZ class integrable probabilistic systems. Recent studies of the SHS6V model and its dynamical version include [OP17, Agg18b, Bor18, BP18, IMS19].
Let us recall the definition of the SHS6V model from [CP16]. Fix , we define the -matrix via
[TABLE]
Here, and is the regularized terminating basic hyper-geometric series defined by
[TABLE]
where we recall the -Pochhammer symbols (here is allowed to be negative) are defined by
[TABLE]
We view as a matrix with row indexed by and column indexed by . Note that the -matrix in (1.4) actually depends on four generic parameters , we suppress the dependence on in the notation of to simplify the notation.
It is straightforward by definition that for (using )
[TABLE]
which means there is no way to transition out of from itself. Therefore, in the following we restrict ourselves to the block with .
When , by straightforward calculation, the -matrix defined above simplifies to
[TABLE]
For the history of the expression (1.4), we remark that more intricate expressions for a quantity similar to the had been known in the context of quantum integrable systems since the work of [KR87]. Relatively compact expressions of became available only in more recent times after the work of [Man14]. [CP16] also provides a probabilistic proof for this expression.
From our perspective, we will think of as the weight associated to a vertex configuration with input lines from south, input lines from west, output lines to the north and output lines to the east see Figure 1. Since we have restricted to , we can have at most vertical lines and horizontal lines in the vertex configuration. Note that due to the indicator in (1.4), all non-zero vertex weights satisfy , a property that we consider as conservation of lines.
In this paper, we always assume the following condition.
Condition 1.1**.**
We take and as we noted before, .
It follows from [CP16] that under Condition 1.1, is a stochastic matrix on . In other words, for any fixed , defines a probability measure on . Although in this paper we will not investigate the range of parameters out of Condition 1.1, it is worth remarking that there are other choices of parameters which make stochastic, a few of them are provided in [CP16, Proposition 2.3].
There are several equivalent ways to define the SHS6V model. In this paper, we view the SHS6V model as a one-dimensional interacting particle system, which follows a sequential update rule. We proceed to give a precise definition of it. Denote by the space of left-finite particle configuration
[TABLE]
where should be understood as the number of particles at position . We define a discrete time Markov process as follows.
Definition 1.2** (left-finite fused SHS6V model).**
For any state , we specify the update rule from state to as follows: Assume the leftmost particle in the configuration is at (i.e. and for all ). Starting from , we update to by setting and randomly choosing according to the probability where . Proceeding sequentially, we update to according to the probability where . Continuing for , we have defined the update rule from to , see Figure 2 for visualization of the update procedure. We call the discrete time-homogeneous Markov process with the update rule defined above the left-finite fused SHS6V model.333Note that in Definition 1.2, although the update from to may never stop as it goes to the right, the process is well-defined since we only care about the sigma algebra generated by for all .
For , we define . For instance,
[TABLE]
We further define for .
Definition 1.3** (left-finite unfused SHS6V model).**
For all state , we specify the update rule at time from state to as follows. Assume the leftmost particle in the configuration is at . Starting from , we update to by setting and randomly choosing according to the probability where . Proceeding sequentially, we update to according to the probability where . Continuing for , we have defined the update rule from to . We call the discrete time-inhomogeneous Markov process with the update rule defined above the left-finite unfused SHS6V model.
Remark 1.4**.**
It is straightforward to check that under Condition 1.1, for all , in (1.5) is a stochastic matrix which transfers to itself.
In this paper, as a notational convention, we always use to denote the fused SHS6V model and to denote the unfused one. The connection between them is specified in the following proposition.
Proposition 1.5** ([CP16], Theorem 3.15).**
Consider the left-finite fused SHS6V model and the left-finite unfused SHS6V model . If in law, then
[TABLE]
By Proposition 1.5, we can construct the SHS6V model with higher horizontal spin () from those with horizontal spin . This procedure is called fusion, which goes back to the work of [KR87]. Thanks to Proposition 1.5, for any left-finite SHS6V model , we can couple it with a left-finite unfused SHS6V model so that . We will extend the definition of unfused SHS6V model in Lemma 2.1 so that it takes value in a larger space of bi-infinite particle configuration (thus extend as well the definition of the fused SHS6V model using the relation ).
For the particle configuration , define
[TABLE]
For the left-finite unfused SHS6V model , we define the unfused height function as
[TABLE]
Note that in the notation of unfused height function, we suppress the underlying process . Similarly, we define the fused height function for the left-finite fused SHS6V model as
[TABLE]
Since , certainly one has for all and ,
We will state our result for the fused height function though we will mainly work with the unfused height function in our proof. In the future, the notation of will often be shortened to .
Having defined (respectively, ) on the lattice, we linearly interpolate it first in space variable then in time variable , which makes (respectively, ) a -valued process. For construction of height functions of the bi-infinite version of the fused or unfused SHS6V model, see Lemma 2.1.
1.3. Result
The main result of our paper shows that the fluctuation of the fused height function converges weakly to the solution of the KPZ equation. Fix , define
[TABLE]
As a matter of convention, we endow the space and with the topology of uniform convergence over compact subsets, and write for the weak convergence of probability laws. We present our main theorem.
Theorem 1.6**.**
Fix b\in\big{(}\frac{I+J-2}{I+J-1},1\big{)}, and , for small , wet and define via . We call this weakly asymmetric scaling. Assume that is nearly stationary with density (see Definition 5.5) and
[TABLE]
then
[TABLE]
where is the Hopf-Cole solution of the KPZ equation
[TABLE]
with initial condition , where the coefficients are given by
[TABLE]
Note that the restriction of in Theorem 1.6 is necessary and sufficient to ensure that Condition 1.1 holds for small enough. In Appendix B, we will demonstrate how our theorem agrees with the non-rigorous KPZ scaling theory used in physics444The KPZ scaling theory is a non-rigorous physics method used to compute the constants (the coefficients of the KPZ equation (1.11) as in our case) arising in limit theorems for the models in the KPZ universality class [KMHH92, Spo12], which has been confirmed in a few cases such as [DT16, Gho17]..
Remark 1.7**.**
In a different setting where (in contrast to Condition 1.1, there is no such that ) and , one can show that is a stochastic matrix on (instead of for our case). In this regime, the SHS6V model allows arbitrary number of particles at each site (instead of at most particles for our case). [CT17] proves the weak universality of the SHS6V model 555In the context of [CT17], the authors prove the weak universality for the higher spin exclusion process defined in [CP16, Definition 2.10], which is equivalent to the SHS6V model after a gap-particle transform. We describe their result in the language of the SHS6V model here. under a different type of weak scaling that corresponds to taking in (1.3). Under this scaling, the number of particles at each site diverges to infinity with rate . This simplifies considerably the control of the quadratic variation of the martingale in the discrete SHE (1.14), which is the main complexity in our analysis.
Remark 1.8**.**
Taking , Theorem 1.6 recovers [CGST18, Theorem 1.1]. We assume in Theorem 1.6 merely due to some technical subtleties we met in Section 7. The proof for needs particular modification and we do not pursue it here.
The proof of Theorem 1.6 will be given in the end of Section 5, as a corollary of Theorem 5.6.
1.4. Method
In this section, we explain the method used in proving Theorem 1.6. Although initially our methods follow [CGST18], rather quickly, we encounter novel complexities that are not present in [CGST18] which require new ideas.
As illustrated in Section 1.2, via fusion, to study the fused SHS6V model, it suffices to work with the unfused version. Similar to [CGST18], the first step is to perform a microscopic Hopf-Cole transform of the SHS6V model (5.6). The existence of the microscopic Hopf-Cole transform is guaranteed by one particle version of the duality (3.8) (which goes back to [CP16, Theorem 2.21]). The microscopic Hopf-Cole transform , which is essentially an exponential version of the unfused height function , satisfies a discrete version of SHE
[TABLE]
Here is an operator which approximates the Laplacian and is a martingale. Owing to the definition of the Hopf-Cole solution to the KPZ equation, Theorem 1.6 is equivalent to showing that the above discrete SHE converges to its continuum version (Theorem 5.6). The proof of Theorem 5.6 reduces to three steps:
1). Showing tightness.
2). Identifying the limit of the linear martingale problem.
3). Identifying the limit of the quadratic martingale problem.
Steps 1) and 2) follow from a similar approach as in [CGST18]. Step 3) is the difficult part; Proposition 6.8 does this by proving a form of self-averaging for the quadratic variation of the martingale . We will focus on discussing the method for proving this self-averaging result in the rest of the section. We remark that other recent KPZ equation convergence results using the Hopf-Cole transform include ASEP- [CST18], Hall-Littlewood PushTASEP [Gho17], weakly asymmetric bridges [Lab17], open ASEP [CS18, Par19].
We will explain what is self-averaging in a moment, but first introduce two tools used in proving it. The first tool is the Markov duality and the second is the exact formula of two particle transition probability of the SHS6V model.
The stochastic six vertex model enjoys two Markov dualities [CP16, Theorem 2.21] and [Lin19, Theorem 1.5]666The Markov duality proved in [Lin19] first appears in [CP16, Theorem 2.23]. In fact [CP16, Theorem 2.23] claims a more general Markov duality for the SHS6V model. In discussions with the authors of [CP16], we recognized a gap in that proof as well as a counter-example to the result when , see [CP19] for detail., which are exploited in proving the self-averaging [CGST18, Proposition 5.6]. The Markov duality in [CP16, Theorem 2.21] also works for the SHS6V model (Proposition 3.6 in our paper), yet it is unknown whether there exists a generalization of [Lin19, Theorem 1.5] for the SHS6V model. [Kua18, Theorem 4.10] discovers a general duality for the multi-species SHS6V model using the algebraic machinery777As a remark, the functional in [Kua18, Theorem 4.10] also serves as the duality functional for a multi-species version of ASEP(q, j), see [CGRS16, Kua17].. At first glance, the duality functional written in [Kua18, Theorem 4.10] takes a rather complicated form, but we only need a two particle version of this duality, in which case the duality functional simplifies greatly (Proposition 3.7 in our paper) and is applicable for proving the desired self-averaging. We remark that this is the first application of [Kua18, Theorem 4.10] as far as we know.
In [BCG16, Theorem 3.6], an integral formula was obtained for general particle transition probability of the stochastic six vertex model via a generalized Fourier theory (Bethe ansatz), using a complete set of eigenfunction of the stochastic six vertex model transition matrix obtained in [BCG16, Theorem 3.4] together with the Plancherel identity [TW08, Theorem 2.1]. [CGST18] applies the steepest descent analysis to a two particle version of this formula to extract a space-time bound, which is the key to control the quadratic variation of the martingale in (1.14).
For the SHS6V model, it is natural to expect that the similar method should apply, since we also have a set of eigenfunctions from [CP16, Proposiiton 2.12] and a generalized Plancherel identity from [BCPS15, Corollary 3.13]. However, the Plancherel identity was originally designed only for and there is a technical issue in extending this identity to which has not been addressed in the existing literatures888[CP16, Proposition A.3] claims the Plancherel identity for can be obtained by analytic continuation of [BCPS15, Corollary 3.13]. After discussions with the authors of [CP16], they agree that there is an issue in this analytic continuation (and the resulting identity) due to poles encountered along the way [CP19]. (see Remark 4.5). Fortunately, we find that when and there are only two particles, such analytic continuation does work, which produces an integral formula for the two particle SHS6V model transition probability (Theorem 4.4). In terms of large contours, the integral formula consists of two double contour integrals and one single contour integral. We find that the single contour integral can be expressed as a residue of one of the double contour integrals. This simplifies our analysis since via certain contour deformation, the single contour integral will be canceled out.
We will analyze (a tilted version of) this integral formula (Corollary 5.3) in Section 7 using steepest descent analysis and obtain a very precise estimate of the (tilted) two particle transition probability defined in (5.20). Compared with the analysis for stochastic six vertex model in [CGST18, Section 6], one difficulty is to find (and justify) the contours for different such that the steepest descent analysis applies. Also in certain cases (Section 7.5) the steepest descent contour can only be implicitly defined (compared with [CGST18, Section 6] where all the steepest descent contour are circles), which complicates our analysis.
Now let us explain what is self-averaging and how these two tools can be applied to prove it. Denote the discrete gradient by . Roughly speaking, the terminology “self-averaging” refers to the phenomena that as
(A) For , the average of over a long time interval of length will vanish.
**(B) ** There exists a positive constant such that the average of over a long time interval of length will vanish.
The proofs of (A) and (B) are given in Lemma 8.2 and Lemma 8.3 respectively, let us make a brief discussion about our strategy here. As we will see in (8.15), under weakly asymmetric scaling,
[TABLE]
where is the density, and is some constant defined in (5.4). Pointwisely, is of the same order as . But (A) tells that after averaging over a long time interval (we will just say ”averaging” afterwards for short), vanishes for , this explains the terminology of “self-averaging”. To prove (A), by the first duality in Lemma 5.2 (which goes back to Proposition 3.6), one is able to write down the conditional quadratic variation in terms of the summation of (a tilted version of) two particle transition probability , i.e. for
[TABLE]
This allows us to move the gradients from and to . We proceed by using a very precise estimate of from Proposition 7.1 (which is proved by making use of the steepest descent analysis of the integral formula of ). Referring to Proposition 7.1, each gradient on \mathbf{V}\big{(}(x_{1},x_{2}),(y_{1},y_{2}),t,s\big{)} gives an extra decay of , which helps us to conclude (A). We remark that for demonstrating (A), our argument is actually simpler than that of [CGST18]. Since we assume , (1.16) holds for all , while in the situation of the stochastic six vertex model , (1.16) holds only for , due to the exclusion restriction (i.e. two particles can not stay in the same site). In fact, [CGST18] needs both of the duality [CP16, Theorem 2.21] and [Lin19, Theorem 1.5] to prove (A).
For (B), there are two tasks: Identifying and proving the self-averaging. These were done simultaneously for the stochastic six vertex model [CGST18]: Note that by (1.15),
[TABLE]
For the stochastic six vertex model, for all , hence . [CGST18, Lemma 7.1] uses this crucial observation to obtain
[TABLE]
By similar method used in demonstrating (A), it is not hard to prove that vanishes after averaging, implying that .
For our case, first we note that with , so the identity obviously fails. We need to find another way to determine and prove the self-averaging. We proceed by first guessing the . Via (1.17), the average of over a long time interval can be approximated by the average of . In Appendix A, we derive a family of stationary distribution of the SHS6V model, which is a product measure , where is a probability measure on indexed by its mean . Starting the SHS6V model from , it is clear that for all . In a heuristic level, one can approximate the average of by that of the \mathbb{E}_{\pi_{\rho}}\big{[}(\widetilde{\eta}_{x+1}(t)-\rho)^{2}\big{]}Z(t,x)^{2}. Under weakly asymmetric scaling, one computes that
[TABLE]
which suggests .
To prove (B) with , note that the second duality in Lemma 5.2 (which goes back to Proposition 3.7) implies
[TABLE]
where approximately999In fact, the functional below is only an approximate version of the duality functional defined in (5.19), we use this approximate version here to avoid extra notations and make our argument more intuitive.
[TABLE]
Note that the expression of is different depending on whether , which is crucial to our proof. Rewriting in terms of the two duality functionals in (1.16) and (1.19)
[TABLE]
It is not hard to show that the second term vanishes after averaging. For the first term above, we combine both of the dualities (1.16), (1.18) and get
[TABLE]
The number of pairs such that compared with is negligible in the summation above so it suffices to study for
[TABLE]
Inserting this expression into the RHS of (1.20) and using the summation by part formula (see (8.39)), we can move the gradient from to . Similar to the argument for (A), applying the estimate in Proposition 7.1 completes the proof of (B).
1.5. Outline
The paper will be organized as follows. In Section 2 we give an equivalent definition of SHS6V model through fusion. At the beginning, we require the existence of a leftmost particle. After that we extend the definition to a bi-infinite version of the SHS6V model (Lemma 2.1), which is the object that we study for the rest of the paper. In Section 3, we introduce two Markov dualities enjoyed by the model. The first one is taken directly from the [CP16, Theorem 2.21]. The second one is a certain degeneration from a general duality in [Kua18, Theorem 4.10]. Section 4 contains the derivation of integral formula for the two point transition probability of the SHS6V model. In Section 5, we define the microscopic Hopf-Cole transform and prove that it satisfies a discrete version of SHE. Due to the definition of the Hopf-Cole solution to the KPZ equation, it suffices to prove that the discrete SHE converges to its continuum version. In Section 6, we prove this result in two steps. First, we establish the tightness of the discrete SHE. Second, we show that any limit point is the solution to the SHE in continuum, assuming the self-averaging property (Proposition 6.8). The last two sections are devoted to the proof of Proposition 6.8. In Section 7, we obtain a very precise estimate for the two point transition probability by applying steepest descent analysis to the integral formula that we obtain in Section 4. In Section 8, we prove Proposition 6.8 using the Markov duality and our estimate of the two point transition probability.
1.6. Notation
In this paper, we denote . denotes the indicator function of an event . We use (respectively, ) to denote the expectation (respectively, probability) with respect to the process or random variable that follow. The symbol represents a circular contour centered at the origin with radius . All contours, unless otherwise specified, are counterclockwise.
1.7. Acknowledgment
The author heartily thanks his advisor Ivan Corwin for suggesting this problem, his encouragement and helpful discussion, as well as reading part of the manuscript. We thank Jeffrey Kuan for the helpful discussion over the degeneration of [Kua18, Theorem 4.10] to the stochastic higher spin six vertex model. We also wish to thank Promit Ghosal and Li-Cheng Tsai, who provided very useful discussions about the technical part of the paper. We are also grateful to two anonymous referees for their valuable suggestions. The author was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program and also received summer support from Ivan Corwin’s NSF grant DMS:1811143.
2. The bi-infinite SHS6V model
The main goal of this section is to extend the definition of the left-finite unfused (fused) SHS6V model in Definition 1.3 (Definition 1.2) to the space of bi-infinite configurations . The motivation of such extension is to include one important class of initial condition called near stationary initial condition as in [BG97]. We will proceed following the idea of [CGST18], which goes back to [CT17]. By fusion (Proposition 1.5), it suffices to extend the left-finite unfused SHS6V model . The extension of the fused version follows easily by taking .
For the extension, the first step is to restate the SHS6V model in a parallel update rule. To this end, we equip the probability space with a family of independent Bernoulli random variables such that
[TABLE]
for , and , recall that .
Treating these Bernoulli random variables as a random environment, we find an equivalent way to define the left-finite unfused SHS6V model, through recursion. Given initial state , define (recall the notation from (1.7)) and recursively for
[TABLE]
It is straightforward to see that is a left-finite unfused SHS6V model and is indeed its height function defined by (1.8).
The recursion (2.2) is equivalent to
[TABLE]
Iterating (2.3) implies
[TABLE]
Note that the summation above is finite. The reason is that since , there exists such that for all , which implies for all .
In light of (2.4), we extend the Definition 1.3 to the space of bi-infinite particle configuration .
Lemma 2.1**.**
For any bi-infinite particle configuration , define the initial height function
[TABLE]
Note that if , defined above coincides with that defined in (1.8). We inductively define the and for via the recursion
[TABLE]
For , the infinite sum in (2.5) converges almost surely and in to a -valued random variable. Furthermore, consider left-finite initial configuration and the height function inductively defined by (2.5) and (2.6), then for all and
[TABLE]
Remark 2.2**.**
It is clear that via (2.5), one can recover the recursion (2.2) since
[TABLE]
In particular, if , the defined in Lemma 2.1 is a left-finite unfused SHS6V model. Therefore, Lemma 2.1 truly extends the scope of Definition 1.3.
Proof of Lemma 2.1.
Define the canonical filtration
[TABLE]
It is not hard to see (via (2.5) and (2.6)) that and are adapted to this filtration.
Let us first justify the convergence of the infinite summation (2.5). To simplify notation, we denote by \mathbb{E}^{\prime}\big{[}\cdot\big{]}=\mathbb{E}\big{[}\cdot\big{|}\mathcal{F}(t)\big{]}. For , denote by
[TABLE]
Observing that for all realization of . Therefore, as , the convergence of implies the almost sure convergence. Note that are independent Bernoulli random variables with mean given in (2.1). As a consequence, there exists constant such that
[TABLE]
Since ,
[TABLE]
Furthermore, note that conditioning on , are all independent, which yields
[TABLE]
Taking expectation on both sides of (2.7), by tower property,
[TABLE]
which implies the convergence of in as .
We proceed to justify
[TABLE]
We prove this by applying induction on . The case is immediately checked. Assuming that we have a proof for , we show that (2.8) also holds for . Note that for all ,
[TABLE]
Since both take value in , we obtain
[TABLE]
Taking , one achieves
[TABLE]
Therefore, in . Since we have assumed (2.8) for , we have
[TABLE]
which completes the induction. ∎
Definition 2.3**.**
We call the defined in Lemma 2.1 the bi-infinite unfused SHS6V model and associate it with the height function defined in Lemma 2.1. We simply define the bi-infinite fused SHS6V model and its height function via
[TABLE]
It is clear that to prove Theorem 1.6, it suffices to work with the bi-infinite unfused SHS6V model. Unless specified otherwise, the SHS6V model now means the bi-infinite unfused SHS6V model . We associate it with the canonical filtration \mathcal{F}(t)=\sigma\Big{(}\vec{\eta}(0),B(s,z,\eta),B^{\prime}(s,z,\eta),0\leq s\leq t-1\Big{)}.
3. Markov duality
One main tool that we rely on to prove Theorem 1.6 is the Markov duality. This is a powerful property which has been found for different interacting particle systems including the contact process, voter model and symmetric simple exclusion process [Lig12, Lig13]. Using Markov duality, Spitzer and Liggett showed that the only extreme translation invariant measures for the SSEP on are the Bernoulli product measure.
In this section, we first state two Markov dualities for the version of left-finite SHS6V model, which comes form [CP16, Theorem 2.21] and [Kua18, Theorem 4.10] respectively. The extension of them to the unfused left-finite SHS6V model is immediate since the transition operators of the model are commute. Finally we explain how to extend these dualities to the bi-infinite unfused SHS6V model constructed in the previous section.
Let us recall the definition of Markov duality in the first place.
Definition 3.1**.**
Given two discrete time Markov processes and (might be time inhomogeneous) and a function , we say that and are dual with respect to if for any and , we have
[TABLE]
The Markov dualities that we are going to present are between the unfused SHS6V model and the -particle reversed unfused SHS6V model location process. To define the latter process, let us first introduce several state spaces.
Definition 3.2**.**
*Recall the space of left-finite particle configuration from (1.6). We likewise define the space of right-finite particle configuration *
[TABLE]
When there are finite number of particles, we restrict and to
[TABLE]
In terms of particle positions, the spaces and are in bijection with
[TABLE]
where denotes the cluster number in , i.e. . should be understood as the location of particles in a non-decreasing order. In particular, we denote by and to be the bijective maps respectively.
Definition 3.3**.**
When , it is clear that Definition 1.2 and Definition 1.3 define the same Markov process. We call it the left-finite SHS6V model. In addition, we call the reversed SHS6V model if is a left-finite SHS6V model.
Since the SHS6V model preserves the number of particles, we can consider SHS6V model with particles as a process on the particle locations.
Definition 3.4**.**
We define the particle SHS6V model location process \vec{x}(t)=\big{(}x_{1}(t)\leq\dots\leq x_{k}(t)\big{)}\in\mathbb{W}_{I}^{k} if (recall the bijective map from Definition 3.2) is the left-finite SHS6V model. We say that is a -particle reversed SHS6V model location process if is a -particle SHS6V model location process. In addition, for , we denote by to be the transition probability from to of the -particle reversed SHS6V model location process. As a matter of convention, could be seen as an operator acting on function in the manner that
[TABLE]
Definition 3.5**.**
We define the -particle unfused SHS6V model location process so that is the left-finite unfused SHS6V model. We say is a -particle reversed unfused SHS6V model location process if is a -particle unfused SHS6V model location process.
Note that for the reversed -particle SHS6V model , we denote by \mathbf{P}_{\overleftarrow{\text{SHS6V}}}\big{(}\vec{x},\vec{y},t,s\big{)} the transition probability from to . Apparently, one has
[TABLE]
It follows from [CP16, Corollary 2.14] (or the Yang-Baxter equation [BP18, Section 3]) that commutes with itself for different values of (i.e. ). Consequently,
[TABLE]
Let us first state the version of Markov duality.
Proposition 3.6** ([CP16], Proposition 2.21).**
For all , the left-finite SHS6V model (Definition 3.3) and -particle reversed SHS6V model location process (Definition 3.4) are dual with respect to the functional
[TABLE]
recall .
In [Kua18], the author discovers a Markov duality for a multi-species version of the SHS6V model. For our application, we explain how to degenerate this result to a two particle SHS6V model duality. Before stating the proposition, let us recall the notation of -deformed quantity
[TABLE]
Proposition 3.7**.**
The left-finite SHS6V model and the two particle reversed SHS6V model location process are dual with respect to
[TABLE]
We remark that there is a misstatement in [Kua18, Theorem 4.10]. The particles in the process and were stated to jump to the left and to the right respectively. However, after discussing with the author, we realize that the right statement is that the particles in jump to the right and those in jump to the left.
Proof.
This is a degeneration from [Kua18, Theorem 4.10]. By taking the species number , the spin parameter for all as well as replacing by , the multi-species SHS6V model considered in [Kua18] degenerates to the SHS6V model (see Section 2.6.2 of [Kua18] for detail). Then Theorem 4.10 of [Kua18] reduces to: The left-finite SHS6V model and the reversed SHS6V model are dual with respect to the functional
[TABLE]
Swapping the role of left and right, which makes the particles in jump to the left and those in jump to the right. Then becomes the left-finite SHS6V model and becomes the reversed SHS6V model. They are dual with respect to the functional
[TABLE]
Assuming has two particles, recall the bijective map (take ) in Definition 3.2, then is the reversed two particle location process. The left-finite SHS6V model and the two particle reversed SHS6V model location process are dual with respect to , where is given by
[TABLE]
In addition, note that
[TABLE]
When , there are at most two values for so that . To make sense of the infinite product in (3.4), one needs to normalize by dividing each factor in the product (3.4) by . After such normalization, it is straightforward via (3.5) that equals the functional in (3.3) up to a constant factor. ∎
We note that the duality functionals in (3.2) and (3.3) do not depend on the parameter . By Markov property and the property that commutes for different value of , it is clear that the same Markov dualities in Proposition 3.6 and Proposition 3.7 apply for the left-finite unfused SHS6V model.
Corollary 3.8**.**
For all , the left-finite unfused SHS6V model (Definition 1.3) and the reversed -particle unfused SHS6V model location process (Definition 3.5) are dual with respect to the functional in (3.2). The left-finite SHS6V model and the two particle reversed unfused SHS6V model location process are dual with respect to the functional in (3.3).
Proof.
Due to Proposition 3.6, we see that for all and ,
[TABLE]
Using Markov property and applying (3.6) repetitively, we see that
[TABLE]
Here, the second equality follows from (3.1). This proves the desired duality with respect to the functional . The duality with respect to the functional follows by a similar argument. ∎
For our application, we like to extend the dualities stated in Proposition 3.6 and Proposition 3.7 to the bi-infinite SHS6V model. Denote by
[TABLE]
Here is the bi-infinite unfused SHS6V model defined in Definition 2.3 and is the associated height function.
Corollary 3.9**.**
For the bi-infinite unfused SHS6V model , for one has
[TABLE]
For (Since , this is equivalent to the condition )
[TABLE]
Proof.
Let us prove (3.8) in the first place. Given initial condition of the bi-infinite unfused SHS6V model , we construct a sequence of left-finite SHS6V model with initial condition . We denote by the associated height function. The first duality in Corollary 3.8 implies that for any
[TABLE]
Let us show that the LHS and RHS of (3.10) approximate those of (3.8) as .
For the approximation of the LHS, as for all , we have . Moreover, in a single time step, may change by at most one, hence for all ,
[TABLE]
Therefore, for fixed and , is uniformly bounded. Via Lemma 2.1, we know that in probability, by conditional dominated convergence theorem, one has
[TABLE]
For the RHS approximation, according to Definition 3.5, when there is only one particle in the reversed SHS6V model location process, it jumps to the left (at time ) as a geometric random variables with parameter . When there are particles, they jump to the left (at time ) as independent geometric random variables with parameter except when one hits another. So there exists constant such that for all
[TABLE]
Denote by , one has
[TABLE]
For fixed , observing that \mathbf{P}_{\overleftarrow{\text{SHS6V}}}\big{(}\vec{y},\vec{x},t,s\big{)} can be written as a -fold convolution of one-step transition probability. The convolution can be expanded into a sum over all trajectories from to . The contribution of each trajectories can be bounded by the product of one-step transition probability, which is upper bounded by the RHS of (3.12). As the particles in the reversed SHS6V model can only jump to the left, the number of the trajectories can be upper bounded by . We obtain
[TABLE]
Furthermore, it is readily verified that under Condition 1.1,
[TABLE]
Using the bounds in (3.11) and (3.13), fix and , we have for all ,
[TABLE]
for some constant . Since in probability, we find that
[TABLE]
Therefore, We conclude (3.8). The proof of (3.9) is similar to (3.8), where we consider instead
[TABLE]
Applying the second duality in Corollary 3.8, we find that
[TABLE]
By taking and using similar approximation, we conclude (3.9). ∎
4. Integral formula for the two particle transition probability
In this section, we give an explicit integral formula for \mathbf{P}_{\overleftarrow{\text{SHS6V}}}\big{(}(x_{1},x_{2}),(y_{1},y_{2}),t,s\big{)} (note that for the rest of the paper, we prefer to swap the order of and in the notation compared with the RHS of (3.9)). Our approach is to utilize the generalized Fourier theory (Bethe ansatz) developed in [BCPS15]. Let us review a few results obtained in [BCPS15] and [CP16] on which we rely to derive the integral formula.
Definition 4.1**.**
For , we define the left and right Bethe ansatz eigenfunction101010Comparing with the original definition for Bethe ansatz function defined in (2.11) and (2.14) of [BCPS15], we reverse the order of components in the vector: We prefer to write instead of .
[TABLE]
where is the permutation group of and
[TABLE]
where denotes the cluster number in , i.e. .
It turns out that are the eigenfunctions of the operator defined in Definition 3.4.
Lemma 4.2** (Proposition 2.12 of [CP16]).**
For all and such that for all , \big{|}\frac{1-w_{i}}{1-\nu w_{i}}\frac{\alpha+\nu}{1+\alpha}\big{|}<1. Then,
[TABLE]
[BCPS15] shows that the left and right Bethe ansatz eigenfunctions enjoy the following bi-orthogonal relation.
Lemma 4.3** (Corollary 3.13 of [BCPS15]).**
For and and ,
[TABLE]
Some notations must be specified here: is a very small circular contour around so as to exclude all the poles of the integrand except . The Plancherel measure is defined as
[TABLE]
where the sum in (4.3) is taken over the partition of , that is to say, with and is the length of the partition . For instance, the partitions of are given by and . We denote by to be number of components that equal in so that . Furthermore, we define
[TABLE]
We are in a position to present our formula.
Theorem 4.4**.**
Assume , for any and , the two point transition probability of reversed SHS6V model admits the following integral formula
[TABLE]
where is a circle centered at zero with large enough radius so as to include all the poles of all the integrands. In addition,
[TABLE]
Note that corresponds to the pole produced by the denominator of and
[TABLE]
denotes the residue of the double contour integral above at the pole .
Proof of Theorem 4.4.
The first step to prove Theorem 4.4 is utilizing the bi-orthogonality of the Bethe ansatz function. Taking in the previous lemma, since the possible partition is either or , we obtain
[TABLE]
Note that according to the previous lemma, (4.6) holds only for , we want to extend this identity to and . This extension can be justified by analytic continuation. Note that the RHS of (4.6) is an analytic function of in a suitable domain which connects and . The reason behind is that after plugging in , there is no new pole of integrand generated inside (Here we use the assumption , this analytic continuation argument is not valid when , see Remark 4.5).
Let us now fix on both side of (4.6) and treat both sides as functions of . We denote by the operator
[TABLE]
Applying the operator on both side of (4.6). For the LHS, it is clear that
[TABLE]
For the RHS, we move inside the integrand, which yields
[TABLE]
Due to Lemma 4.2 (note that is a small circle around , hence satisfy the condition of Lemma 4.2),
[TABLE]
We name the first term on the RHS of (4.7) and the second term ,
[TABLE]
We compute in the first place. In the integrand of (4.8), the function is a symmetrization of
[TABLE]
Furthermore, all other terms of the integrand (4.8) are symmetric function of . In addition, we are integrating along the same contour, this allows us to desymmetrize the integrand
[TABLE]
We readily calculate
[TABLE]
Replacing the terms and in the integrand of (4.10) by the RHS of (4.11) and (4.12), one sees that
[TABLE]
For the second equality above, we changed \big{(}\frac{1-w_{i}}{1-\nu w_{i}}\big{)}^{y_{3-i}-x_{3-i}} to \big{(}\frac{1-w_{i}}{1-\nu w_{i}}\big{)}^{y_{i}-x_{i}}, due to the symmetry of .
We proceed to compute , by a straightforward calculation
[TABLE]
Inserting these expressions into the integrand of (4.9) gives
[TABLE]
A crucial observation is that one can verify directly
[TABLE]
Note that \mathbf{P}_{\overleftarrow{\text{SHS6V}}}\big{(}(x_{1},x_{2}),(y_{1},y_{2}),t,s\big{)}=I_{1}+I_{2}, using (4.13) and (4.14) one has
[TABLE]
Recall that for all , we can simplify the telescoping product in the integrand via
[TABLE]
Furthermore, referring to the expression (4.1) and (4.5), we notice that . Thereby,
[TABLE]
Lastly, we transform the small circle surrounding into the big circle via a change of variable
[TABLE]
By the following relations
[TABLE]
we obtain
[TABLE]
This concludes the proof of Theorem 4.4. Note that we change the sign in front of the residue from (4.15) to (4.16). This is due to the fact that, before employing the change of variable, the set of the poles lies outside the -contour , while after the change of variable, the set of the pole lies inside the -contour , since is chosen to be sufficiently large. ∎
Remark 4.5**.**
We remark that our argument in proving that (4.6) holds for and does not work when . The reason is as follows: Note that the factor in the integrand of (4.6) gives a pole for the -contour at . Before the substitution of , this pole lies outside the contour . Yet after substituting , the pole becomes , which runs inside the contour , hence the argument of analytic continuation fails. This issue is also addressed in [BCPS19], when the authors tried to reproduce the integral formula for the particle ASEP transition probability (which first appears in [TW08, Theorem 2.1]) via analytic continuation of (4.2). For a similar reason, our method does not yield the general particle transition probability formula of the SHS6V model.
5. Microscopic Hopf-Cole transform and SHE
In this section, we first define the microscopic Hopf-Cole transform , which is an exponential transform of the height function . Using version of duality of (3.8), it turns out that satisfies a discrete version of SHE. As the Hopf-Cole solution to the KPZ equation is the logarithm of the mild solution of the SHE, this reduces the proof of Theorem 1.6 to proving that converges to the solution of SHE. We will derive two Markov dualities for in Lemma 5.2, as a tilted version of (3.8). This will be used in the proof of self-averaging property Proposition 6.8.
5.1. Microscopic Hopf-Cole Transform
We first study a one particle version of the unfused SHS6V model location process (Definition 3.5). When there is only one particle, it performs a random walk where are independent (but not same distributed) -valued random variables with distribution
[TABLE]
By tilting and centering with respect to \mathbb{E}\big{[}q^{\rho R^{\prime}(k)}\mathbf{1}_{\left\{R^{\prime}(k)=\cdot\right\}}\big{]}, we define a tilted random walk , where are independent valued with distribution111111The tilted and centered random walk provides the heat kernel for the discrete SHE (5.7) satisfied by the microscopic Hopf-Cole transform (5.6), which is an exponential transform of the LHS of (1.10).
[TABLE]
Here, \lambda(k)=\big{(}\mathbb{E}\big{[}q^{\rho R(k)}\big{]}\big{)}^{-1} is the normalizing parameter and is the centering parameter which makes \mathbb{E}\big{[}R(k)]=0. Under straightforward calculation, we see that
[TABLE]
We remark that (respectively ) are periodic in the sense that (respectively ). Denote by
[TABLE]
It can be verified that the parameter , defined in (1.9) satisfies
[TABLE]
hence, one has
[TABLE]
We define the microscopic Hopf-Cole transform for as
[TABLE]
For , we set \mathsf{p}(t,s,x):=\mathbb{P}\big{(}X(t)-X(s)=x\big{)}. Denote by the convolution
[TABLE]
We set
[TABLE]
We sometimes call the flux, since it records the number of particles (either zero or one) that move across the position between time and . Now we present the discrete SHE satisfied by the microscopic Hopf-Cole transform of the unfused SHS6V model.
Proposition 5.1**.**
For and , satisfies the following discrete SHE
[TABLE]
where
[TABLE]
Furthermore, is a martingale increment, i.e. \mathbb{E}\big{[}M(t,x)\big{|}\mathcal{F}(t)\big{]}=0. The conditional quadratic variation of equals
[TABLE]
where
[TABLE]
Proof.
We first show that is a martingale increment. Note by (5.7),
[TABLE]
Taking in the duality (3.8), one has
[TABLE]
Hence,
[TABLE]
which implies \mathbb{E}\big{[}M(t,x)\big{|}\mathcal{F}(t)\big{]}=0.
We turn to justify (5.8). Note that by (5.6)
[TABLE]
Since ,
[TABLE]
Combining with (5.12) gives
[TABLE]
which gives the desired equality.
We turn our attention to (5.9). Define the short notation E^{\prime}\big{[}\cdot\big{]}:=\mathbb{E}\big{[}\cdot\big{|}\mathcal{F}(t)\big{]} and write , to be the corresponding conditional variance and covariance. We assume without loss of generosity and use shorthand notation . Owing to (5.14),
[TABLE]
Define
[TABLE]
where are defined in (2.1). Since are all independent, due to the expression (2.5) of provided by (2.5), it is straightforward that conditioning on , are independent with . Furthermore, (2.5) implies
[TABLE]
By the independence, we see that
[TABLE]
Referring to (5.16),
[TABLE]
Inserting this into the RHS of (5.17), we find that
[TABLE]
Here, the last equality follows from the fact . Furthermore, due to (5.13),
[TABLE]
Inserting this into the RHS of (5.18) yields
[TABLE]
Using the fact , we obtain
[TABLE]
Combining with (5.15), we arrive at the desired (5.9). ∎
For , define
[TABLE]
We consider a tilted version of the duality functional in (3.7), for , define
[TABLE]
We further define for and ,
[TABLE]
Observe that is a tilted version of , thus it is clear that it inherits the two dualities stated in Corollary 3.9.
Lemma 5.2**.**
For and ,
[TABLE]
Proof.
We use the shorthand notation . Referring to (5.6),
[TABLE]
Using Corollary 3.9, we have
[TABLE]
Inserting this into the RHS of (5.23), via a straightforward computation, we conclude (5.21). The second duality (5.22) follows from a similar argument, we do not repeat here. ∎
The following corollary follows from Theorem 4.4.
Corollary 5.3**.**
For all and , we have
[TABLE]
where is a circle centered at zero with a large enough radius so as to include all the poles of the integrands, is defined in (4.5) and
[TABLE]
Proof.
Note that the integral formula for is given by (4.4), referring to (5.20), we find that
[TABLE]
We refer to the context of Theorem 4.4 for the notation. Multiplying the constant \big{(}\frac{\hat{\lambda}(t)}{\hat{\lambda}(s)}\big{)}^{2}q^{\rho\left(x_{1}+x_{2}-y_{1}-y_{2}+2\hat{\mu}(t)-2\hat{\mu}(s)\right)} to each term inside the square bracket above and applying change of variable readily yield the desired formula. ∎
5.2. The SHE
Consider the KPZ equation with parameter and given in (1.12) and (1.13),
[TABLE]
As mentioned in Section 1.1, via formally applying Hopf-Cole transform, we say that is a Hopf-Cole solution of (5.29) if
[TABLE]
where is a mild solution of the SHE
[TABLE]
in the sense that it satisfies the following Duhamel integral form
[TABLE]
where is the heat kernel. The stochastic heat equation has a unique mild solution , see [Cor12] and references therein.
We recall the weakly asymmetric scaling for the SHS6V model stated in Theorem 1.6:
[TABLE]
Such scaling corresponds to taking and keeping unchanged in (1.3). Note that all parameters in the SHS6V model rely on the generic parameters , since under weakly asymmetry scaling, are all fixed and , the evolution of the entire model depends on . As we will let go to zero, it suffices to consider all small enough, which means that we only consider for some generic but fixed threshold .
Lemma 5.4**.**
Under weakly asymmetric scaling (5.30), we have the following asymptotics near
[TABLE]
As notational convention, we denote to be a generic quantity such that .
Proof.
For every , we have , and , where are fixed. The relation of and with is implied by (5.2) and (5.3) The verification of the above asymptotic is then straightforward. ∎
To highlight the dependence on under weakly asymmetric scaling, we denote by the microscopic Hopf-Cole transform . Note that presently is only defined for and , we extend to be a -valued process by first linearly interpolating in , then in . This is slightly different from exponentiating the interpolated height function . Nevertheless, under the weak asymmetric scaling , it is straightforward to see that the difference between these two interpolation schemes is negligible as .
As a notational convention, we write \big{\|}X\big{\|}_{p}:=(\mathbb{E}|X|^{p})^{\frac{1}{p}} for . Following the work of [BG97], we define the near stationary initial data for the unfused/fused SHS6V model.
Definition 5.5**.**
Fix , we call the initial data (equivalently ) near stationary with density if for any and , there exists constant and such that for all
[TABLE]
holds for small enough.
Theorem 5.6**.**
Under weakly asymmetric scaling, assuming that is near stationary with density and for some -valued process
[TABLE]
then
[TABLE]
where is the mild solution to the SHE
[TABLE]
with initial condition .
As a consequence of the preceding theorem, we prove Theorem 1.6.
Proof of Theorem 1.6.
Via the discussion in Section 5.2, solves the KPZ equation
[TABLE]
One has by (5.7),
[TABLE]
By Theorem 5.6 and continuous mapping theorem, we obtain
[TABLE]
In other words,
[TABLE]
Note that we have (although in fact, they only equal on the lattice due to different linear interpolation scheme, but it is obvious that the difference between them is negligible). Moreover, via (5.5)
[TABLE]
Therefore, replacing the time variable with in (5.32),
[TABLE]
where . It is straightforward to check that satisfies the KPZ equation
[TABLE]
which concludes the proof of Theorem 1.6. ∎
6. Tightness and proof of Theorem 5.6
In this section, we prove Theorem 5.6 assuming Proposition 6.8, whose proof is postponed to Section 8. First of all, we prove the tightness of , which indicates that as , converges weakly along a subsequence. To identify the limit as well as proving the convergence of the entire sequence, we appeal to the martingale problem of SHE that was first introduced in the work of [BG97]. Using approximation from the microscopic SHE (5.7) to the SHE in continuum, we show that any subsequential limit of satisfies the same martingale problem, hence is the mild solution of SHE.
Hereafter, we always assume that we are under weakly asymmetric scaling (5.30). In general, we will not specify the dependence of parameters on . We will also write ,, etc. when we do want to emphasize the dependence. The dependence on , , will not be indicated as they are fixed.
For the ensuing discussion, we will usually write for constants. We might not generally specify when irrelevant terms are being absorbed into the constants. We might also write when we want to specify which parameters the constant depends on. We say “for all small enough” if the referred statement holds for all for some generic but fixed threshold that may change from line to line.
6.1. Moment bounds and tightness
The goal of this section is to prove the following Kolmogorov-Chentsov type bound for the microscopic Hopf-Cole transform.
Proposition 6.1**.**
Assume that we start the SHS6V model with near stationary initial data with density . Given , , . There exists positive constants , such that
[TABLE]
for all and .
We immediately deduce the tightness of once we have the moment bound above.
Corollary 6.2**.**
The law of -valued process is tight.
Proof.
(6.1), (6.2) and (6.3) indicate that with large probability is uniformly bounded, uniformly spatially and uniformly temporally Hölder continuous. Applying Arzela-Ascoli theorem together with Prokhorov’s theorem [Bil13] yields the desired result. ∎
For the proof of Proposition 6.1, we will basically follow the framework developed in [CGST18]. Let us begin with a technical lemma which will be frequently used for the rest of the paper.
Lemma 6.3**.**
Fix , for any , there exists such that for all and , there exists such that for all positive , and , the following inequality holds121212Here can be any positive constant, though for application, the choice of it usually depends on the value of .
[TABLE]
Proof.
Take , for and arbitrary , due to , one has
[TABLE]
holds for , where is is to be chosen small enough. Thereby,
[TABLE]
Here, the last inequality follows from
[TABLE]
Thus, we conclude the lemma. ∎
The following estimate for the one particle transition probability will be useful in proving Proposition 6.1.
Lemma 6.4**.**
For any and , there exists constant (depending on ) such that
[TABLE]
for small enough and .
Proof.
The proof is more or less analogous to [CGST18, Lemma 5.1]. We first claim that admits the following integral formula
[TABLE]
where , are defined in (5.25) and (5.26) respectively and is large enough so that the circle includes all the singularities of the integrand. This claim can be proved by observing
[TABLE]
This implies
[TABLE]
Via Fourier inversion formula, we have
[TABLE]
In Section 7, we will obtain an upper bound of by applying steepest descent analysis to the integral formula above and we use this upper bound here in advance. Referring to (7.21), by taking , we obtain for all , there exists positive constant such that for small enough
[TABLE]
which gives (i). Using (6.6) together with Lemma 6.3 gives (ii)
[TABLE]
For (iii), we see that
[TABLE]
For the second inequality above, we used the inequality
[TABLE]
Finally, to prove (iv), one has by (7.24) (taking )
[TABLE]
Summing the above equation over (assuming with out loss of generosity that ), we obtain
[TABLE]
If we bound each term in the geometric sum by , we have \big{|}\mathsf{p}(t,s,x)-\mathsf{p}(t,s,x^{\prime})\big{|}\leq\frac{C}{t-s+1}|x^{\prime}-x|. In addition, we can bound the geometric sum by
[TABLE]
which implies
[TABLE]
Thereby,
[TABLE]
which concludes the proof of (iv). ∎
Recall the discrete SHE in Proposition 5.1
[TABLE]
Iterating (6.7) for times yields
[TABLE]
where the martingale equals
[TABLE]
To estimate , it suffices to estimate and respectively. In general, the former one is easier to bound due to Lemma 6.4, while controlling the latter one is much harder. Following the style of [CGST18], to estimate , we need to establish the following two lemmas, which are in analogy with Lemma 5.2 and Lemma 5.3 of [CGST18].
Let denote the set of the partitions into intervals of or elements. Here, the interval refers to the set of form , . For example,
[TABLE]
For and , we define .
Lemma 6.5**.**
Fix , for all and , we have
[TABLE]
Proof.
[CGST18, Lemma 5.2] proved this inequality for . When , the proof is almost the same. Let us denote by \mathbb{E}^{\prime}\big{[}\cdot\big{]}=\mathbb{E}\big{[}\cdot\big{|}\mathcal{F}(t)\big{]} and
[TABLE]
Due to (2.7), there exists such that
[TABLE]
This gives bound similar to (5.10) in [CGST18, Lemma 5.2]. The rest of the proof is the same as in [CGST18, Lemma 5.2], we do not repeat it here. ∎
Lemma 6.6**.**
Fix , recall the martingale increment from (5.7) and let be a deterministic function defined on and . Write , we have
[TABLE]
Proof.
Using the previous lemma, the proof is the same as the one appeared in [CGST18, Lemma 5.3]. ∎
Have prepared the preceding lemmas, we proceed to prove Proposition 6.1. Here we use a slightly different approach compared with the proof of the moment bounds in [CGST18, Proposition 5.4].
Proof of Proposition 6.1.
Recall that is defined on through linear interpolation. It suffices to prove the theorem for the lattice and . Generalization to continuum follows easily.
Let us begin with proving (6.1). We have by (6.8)
[TABLE]
Using , we get
[TABLE]
For the first term on RHS of (6.10), by Cauchy-Schwarz inequality,
[TABLE]
For the second term \big{\|}Z_{mg}(t)\big{\|}_{2n}^{2}, by (6.9)
[TABLE]
Applying Lemma 6.6, there exists a constant so that
[TABLE]
where the last inequality follows from Theorem 6.4 (i).
Replacing the RHS of (6.10) by upper bound obtained in (6.11) and (6.12), we obtain
[TABLE]
Define the set for . Iterating (6.13) yields
[TABLE]
where . Following Lemma 6.4, we bound
[TABLE]
For the second term on the RHS of (6.14), note that via integral approximation, we readily see that
[TABLE]
where is the Gamma function. Combining (6.15) and (6.16) yields
[TABLE]
Note that (since ), as the growth rate of is much faster than that of , the infinite series in the parentheses above converge, which concludes (6.1).
The proof for (6.2) and (6.3) relies on (6.1). We proceed to prove (6.2), denote by
[TABLE]
Using (6.8) (subtract from ) , we have
[TABLE]
where
[TABLE]
It is straightforward that
[TABLE]
By the definition of the near stationary initial data (Definition 5.5), for , there exists such that
[TABLE]
Further applying Theorem 6.4 (ii), one has
[TABLE]
We conclude that
[TABLE]
To bound \big{\|}Z_{mg}^{\nabla}(t)\big{\|}_{2n}, we appeal to Lemma 6.6. Note that due to Lemma 6.4 (iv),
[TABLE]
Applying Lemma 6.6 to (6.17) implies
[TABLE]
Owing to Theorem 6.4 (i), we observe that
[TABLE]
Consequently,
[TABLE]
We conclude (6.2) via combining (6.18) and (6.19).
Finally, we justify (6.3), we have
[TABLE]
where . Similar to the previous proof, we have
[TABLE]
For the first term on the RHS of (6.20), we apply (6.2) and Lemma 6.4 (iii), for any ,
[TABLE]
For the second term, invoking Lemma 6.6 gives
[TABLE]
Combining (6.20)-(6.21), we obtain \big{\|}Z(t,x)-Z(t^{\prime},x)\big{\|}_{2n}\leq C(\epsilon^{2}(t-t^{\prime}))^{\frac{a}{2}}e^{u\epsilon|x|}. We complete the proof of Proposition 6.1. ∎
Having shown the tightness of , to prove Theorem 5.6, it suffices to show that any limit point of is the mild solution to the SHE (5.31). This is the goal of the Section 6.2 and Section 6.3, where we will formulate the notion of “solution to the martingale problem” (which is equivalent to the mild solution) and prove that any limit point of satisfies the martingale problem.
6.2. The martingale problem
We recall the martingale problem of the SHE from [BG97].
Definition 6.7**.**
We say that a -valued process is a solution of martingale problem of the SHE (5.31)
[TABLE]
with initial condition if in distribution and
(i) Given any , there exists such that
[TABLE]
(ii) For any test function ,
[TABLE]
is a local martingale.
(iii) For any test function ,
[TABLE]
is a local martingale.
[BG97, Proposition 4.11] proves the the solution to the martingale problem is also the weak solution (equivalently, the mild solution) to the SHE. Moreover, they show that there is a unique such solution.
To prove Theorem 5.6, it suffices to prove that any limit point of satisfies (i), (ii), (iii). We will do it in the next section. The main difficulty arises for justifying the quadratic martingale problem (iii), we need the following proposition, whose proof is postponed to Section 8.
Proposition 6.8**.**
For , define
[TABLE]
Start the unfused SHS6V model from near stationary initial condition, for given , there exists constant and such that (recall the expressions and from (5.10))
[TABLE]
for all , and small enough.
Remark 6.9**.**
In (6.23), we compensate the space variable by to ensure that .
6.3. Proof of Theorem 5.6
The entire section is devoted to the proof of Theorem 5.6. As we mentioned earlier, due to the tightness obtained in Proposition 6.1, if suffices to prove that for any limit point of satisfies the martingale problem. The proof is accomplished once we verify (i), (ii), (iii) for .
For the ensuing discussion, we denote by to be a generic process (which may differ from line to line) satisfying for all fixed
[TABLE]
We start by verifying (i). Due to (6.1) and , by Skorohod representation theorem and Fatou’s lemma, (i) holds.
We continue to prove (ii). To show that is a local martingale, we consider a discrete analogue. Define
[TABLE]
Due to Proposition 5.1, is a -martingale increment, which implies is a -martingale.
Define . By (5.7),
[TABLE]
we obtain
[TABLE]
Summing (6.25) over yields
[TABLE]
Recall that is the random variable defined in (5.1), as usual we put on the subscript to emphasize the dependence. Note that,
[TABLE]
By Taylor expansion
[TABLE]
whereby (6.26) becomes
[TABLE]
Furthermore, we have
[TABLE]
In the last line, we used Lemma 5.4 to get asymptotics. Denote by
[TABLE]
Then
[TABLE]
Note that is a periodic sequence with period , by the time regularity of in (6.3), we can replace by
[TABLE]
as defined in (1.12). Consequently,
[TABLE]
Since \lim_{\epsilon\downarrow 0}\sup_{t\in[0,\epsilon^{-2}T]\cap\mathbb{Z}}\big{\|}\mathcal{E}_{\epsilon}(t)\big{\|}_{2}=0, by a standard discrete to continuous argument from the martingale to , we conclude that is a local martingale.
We finish the proof of (iii) based on Proposition 6.8. Similar to what we did in proving (ii), we want to find a discrete approximation of . This is given by . Referring to (6.24), the martingale possesses the quadratic variation
[TABLE]
where the last equality follows from Proposition 5.1. Since , there exists a constant such that
[TABLE]
Consequently, the expression (6.28) is well-approximated with the corresponding term replaced by , which yields
[TABLE]
Here, in the third equality we used . In the last equality, using Lemma 5.4 for asymptotics expansion of , one has
[TABLE]
Using Proposition 6.8, we replace the term in (6.29) with ,
[TABLE]
Using again the time regularity of in (6.3), we conclude that
[TABLE]
where
[TABLE]
as defined in (1.13). Via a standard discrete to continuous argument from the martingale to , we conclude that is a local martingale. Since we have proved that for any limit point of , it satisfies (i), (ii), (iii) in Definition 6.7, this concludes the proof of Theorem 5.6.
7. Estimate of the two particle transition probability
In this section, we prove a space-time estimate for the (tilted) two particle transition probability , using the integral formula provided in Corollary 5.3. This technical result is crucial to the proof of Proposition 6.8.
Recall from Corollary 5.3 that
[TABLE]
where is a circle centered at zero with a large enough radius so as to include all the poles of the integrand, is defined in (4.5) and the functions in the integrand above are defined respectively in (5.25) - (5.28). We put in the notation of and other functions to emphasize the dependence on under the weakly asymmetry scaling.
We define the discrete gradients
[TABLE]
Furthermore, we define the mixed discrete gradient
[TABLE]
We define the -Weyl chamber (which is understood with respect to whichever gradient is taken) to be
[TABLE]
We remark that \mathbf{V}_{\epsilon}\big{(}(x_{1},x_{2}),(y_{1},y_{2}),t,s\big{)} is defined only for and . In the definition of -Weyl chamber, when , the corresponding -Weyl chamber is exactly where the quantities , or are well defined. But for , we require , which is stronger than (where is well defined). The motivation of this requirement is to ensure that (7.9) holds.
The following result is the main technical contribution of our paper.
Proposition 7.1**.**
*For all fixed , there exists positive constant such that for small enough and
For all and ,*
[TABLE]
* For all in the -Weyl chamber,*
[TABLE]
(c) For all and ,
[TABLE]
It is helpful to divide the proof of Proposition 7.1 depending on whether the time increment is large enough. More precisely, we use the phrase is large enough if the referred statement holds for all , where is some generic time threshold which may change from line to line (depend on and , but does not depend on ). Note that this is not to be confused with the global assumption , which implies .
Given arbitrary fixed , let us first prove the proposition for .
Proof of Proposition 7.1 for .
According to Lemma 5.4,
[TABLE]
here we used the condition in (5.30). Taking in (3.13) yields
[TABLE]
where . So there exists such that for small enough and all such that
[TABLE]
Referring to the relation (5.20) between and . By along with (7.6), there exists s.t.
[TABLE]
Consequently, we can take and in (7.3) large enough such that for ,
[TABLE]
For the gradients, let us consider for example. Note that
[TABLE]
Using the same argument as above, there exists constant and such that for all satisfying ,
[TABLE]
which gives the desired bound for \nabla_{x_{1}}\mathbf{V}_{\epsilon}\big{(}(x_{1},x_{2}),(y_{1},y_{2}),t,s\big{)}. The argument for the gradient and is similar. ∎
Having proved Proposition 7.1 for , it suffices to prove the same proposition for large enough. In other words, we need to show that there exists such that the proposition holds for . We decompose (7.1) by
[TABLE]
where
[TABLE]
Referring to (4.5), equals as long as . It is straightforward that for in the -Weyl chamber (7.2),
[TABLE]
In addition, for and ,
[TABLE]
Note that under weakly asymmetric scaling,
[TABLE]
which implies that is uniformly bounded for small enough, This being the case, to prove Proposition 7.1 for large enough, it suffices to prove the same result for and respectively.
Proposition 7.2**.**
*For all , there exists positive constant and such that for small enough and satisfying
for all , *
[TABLE]
* For all in the -Weyl chamber,*
[TABLE]
(c) For all and ,
[TABLE]
Proposition 7.3**.**
*For all , there exists positive constant and such that for small enough such that ,
for all and ,*
[TABLE]
* For all in the -Weyl chamber,*
[TABLE]
(c) For all and ,
[TABLE]
The reader might notice that in Proposition 7.3, we write on the RHS exponents (compared with in Proposition 7.1). This in fact yields a stronger upper bound since by and , one always has
[TABLE]
Hence, combining Proposition 7.2 and Proposition 7.3, we conclude Proposition 7.1.
7.1. Estimate of
In this section, we will prove Proposition 7.2. Referring to (6.4),
[TABLE]
where is large enough so that encircles all the poles of the integrand. Therefore, from (7.7) we have
[TABLE]
To estimate \mathbf{V}_{\epsilon}\big{(}(x_{1},x_{2}),(y_{1},y_{2}),t,s\big{)}, it suffices to analyze . Referring to the expression (5.25) and (5.26),
[TABLE]
Define the set of poles of the integrand in (7.10) to be , it is clear that
[TABLE]
Due to Lemma 5.4,
[TABLE]
Therefore, there exists such that for small enough
[TABLE]
To extract the spatial decay of , we deform the contour of from to where
[TABLE]
Note that when is large enough, is close to , thus deforming the contour from to , we do not cross the poles in the integrand. We parametrize by and get
[TABLE]
We want to bound each terms that appear in the integrand above. Note that by (7.15), .
To estimate , referring to (7.13), is a product of up to terms (since ). For each term, by Lemma 5.4
[TABLE]
The singularities in (7.16) lie strictly inside the unit disk. Since is close to when is large, for small enough and large enough, there exists constant such that for and
[TABLE]
which implies
[TABLE]
Consequently,
[TABLE]
We expect to extract the temporal decay from the integral above. To this end, we need to the following lemma.
Lemma 7.4**.**
There exists positive constants , such that for
[TABLE]
holds for small enough and large enough .
As a remark, we see from (7.12) that the function is not globally analytic due to the factor ( is not an integer), but it is analytic in a neighborhood of . Furthermore, \big{|}\mathfrak{D}_{\epsilon}(z)\big{|} is a continuous function in a neighborhood of the unit circle.
Proof of Lemma 7.4.
We only prove Lemma 7.4 for , the argument for is similar. By writing \big{|}\mathfrak{D}_{\epsilon}(z(\theta))\big{|}^{t-s}=e^{(t-s)\text{Re}\,\log\mathfrak{D}_{\epsilon}(z(\theta))}, it suffices to show that there exists positive constants such that for small enough and large enough
[TABLE]
where denotes the real part of a complex number .
We divide our proof into three cases. It suffices to show
- •
- •
( small): There exists s.t.
[TABLE]
- •
( large): There exists such that \big{|}\mathfrak{D}_{\epsilon}(\mathsf{u}(t-s,\beta)e^{\mathbf{i}\theta})\big{|}<1-\delta for .
The proof for the first and second bullet point are done by using the local property of near (applying Taylor expansion). Let be a small neighborhood around such that is analytic inside .
**(): ** We write into terms of a telescoping product
[TABLE]
By (6.5), we see that
[TABLE]
thus
[TABLE]
Referring to (6.27),
[TABLE]
where is given by (1.12). The above discussion implies that
[TABLE]
Moreover, there exists constant such that uniformly for and small enough,
[TABLE]
Since , we see that for large enough. Thus, we taylor expand around and get
[TABLE]
which justifies the first bullet point.
( small): Consider the function , we calculate for
[TABLE]
Given these properties, we taylor expand at , there exists such that
[TABLE]
In conjunction with (which is shown by (7.19)), we conclude the second bullet point.
( large): We set
[TABLE]
Referring to the expression of in (7.12) and using Lemma 5.4, one has
[TABLE]
The convergence is uniform in an open neighborhood of unit circle. Thereby,
[TABLE]
As a result, we conclude the third bullet point as long as we verify the following steepest descent condition
[TABLE]
To prove (SD.), we compute
[TABLE]
In the last step, we used the condition . ∎
Having proved Lemma 7.4, we proceed to finish the proof of Theorem 7.2.
Proof of Theorem 7.2.
Due to Lemma 7.4,
[TABLE]
This being the case, by (7.18) we readily see that
[TABLE]
Incorporating this bound into (7.11) concludes Theorem 7.2 part (a).
For the gradient, notice that one has
[TABLE]
The proof for gradients , is similar to that for by symmetry. It suffices to analyze
[TABLE]
By the fact \big{|}z_{1}(\theta_{1})-1\big{|}=\big{|}e^{\pm\frac{\beta}{\sqrt{t-s+1}}+\mathbf{i}\theta_{1}}-1\big{|}\leq C(\frac{1}{\sqrt{t-s+1}}+|\theta_{1}|), we conclude
[TABLE]
where the last inequality follows by a change of variable . Incorporating this bound into (7.22) and (7.23), we conclude the Theorem 7.2 (b), (c). ∎
7.2. Estimate of , an overview.
Recall from (7.8) that
[TABLE]
We study the double contour integral in (7.25). Recall from (5.27) and (5.28) that
[TABLE]
which produces a pole at where
[TABLE]
Referring to (7.14), the other poles of the integrand belong to for some .
We say the contour is admissible if
[TABLE]
where the distance between a point and a set is define by . Figure 3 below gives several graphical examples of admissible and not admissible contours.
Define
[TABLE]
Note that
[TABLE]
Note that , from above we have: For large enough and small enough, if is admissible, deforming the -contour from to will cross the pole for all . Moreover, such deformation does not cross any other poles in . Therefore,
[TABLE]
In practice, we deform the -contour to some contour which depends on both and so that it is admissible for large enough and small enough.
Assuming that we have deformed -contour to , which is admissible. The next step is to deform the -contour. Note that given , generates a pole at ( is the inverse of )
[TABLE]
We consider three potential radius
[TABLE]
where is a constant which is irrelevant with the current discussion. We deform -contour from to , where
[TABLE]
In other words, if the pole lies outside , we choose -contour to be a circle with radius . If the pole lies inside , we choose -contour to be circle with radius . It is clear we always have for large enough that
[TABLE]
Referring to the expression of (7.26), we find that
[TABLE]
We set
[TABLE]
From preceding discussion, we decompose , where
[TABLE]
Note that we integrate under the indicator , which arises in the case that deforming the -contour from to crosses the pole .
We want to perform the steepest descent argument for and , similar to what we have done in Section 7.1. More precisely, as and , converges to some fixed contour .131313We define the distance of two contours to be \text{dist\,}\big{(}\Gamma_{1},\Gamma_{2}\big{)}=\sup_{x\in\Gamma_{1},y\in\Gamma_{2}}\big{(}d(x,\Gamma_{2})\vee d(y,\Gamma_{1})\big{)}. We say a sequence of contours converges to if We set
[TABLE]
Recall from (7.20) that
[TABLE]
and set
[TABLE]
Note that
[TABLE]
We require the contour satisfying the steepest descent condition.
[TABLE]
As we see from (SD.) that if we take , holds. However, does not hold. In truth, Figure 4 indicates the region where and for and . We see that lies fully inside , but partially outside .
Set \mathcal{M}=\{\big{|}z-\frac{1}{I+1}\big{|}=\frac{I}{I+1}\}, the following lemma says that the satisfies the steepest descent condition (7.34).
Lemma 7.5**.**
We have
[TABLE]
Proof.
Parametrize by , we compute
[TABLE]
where in the first line we used the fact and in the last line we used , note that when and , we have
[TABLE]
which concludes the last inequality.
For , note that
[TABLE]
A crucial observation is that \big{|}z-\frac{1}{I+1}\big{|}=\frac{I}{I+1} implies
[TABLE]
which can be verified by inserting . Consequently, we see that
[TABLE]
where
[TABLE]
We claim that , which implies . This claim is justified by showing
[TABLE]
Therefore, by and (7.35)
[TABLE]
which concludes our proof. ∎
We need to consider the following modification of
[TABLE]
where is some positive real number.
Lemma 7.6**.**
There exists such that for all , one has
[TABLE]
Proof.
The proof of this lemma uses similar techniques which appear in [CGST18, Lemma 6.4]. By straightforward computation, one finds that
[TABLE]
Here, is given by (1.12). We taylor expand and around and get
[TABLE]
Notice that in the vertical direction where , is negative. This implies that
[TABLE]
where is a hourglass region centered at one, with fixed. For , due to and Lemma 7.5, we find that there exists a small , such that for
[TABLE]
Combining this with (7.36) concludes the proof of Lemma 7.6. ∎
We fix a constant and set . From our discussion above, is admissible and satisfies (SD).
To prove Proposition 7.3, we need to choose our contour such that it controls both and . The choice will depend on the sign of and . We need to discuss separately for each of the following cases
(i): case: and ,
(ii): case: and ,
(iii): case: and .
Note that we don’t need to consider the case where and , since it contradicts our condition and .
7.3. Estimate of , the () case
In this case we shrink the -contour from to
[TABLE]
It is clear that for large enough, is admissible. Consequently, we have
[TABLE]
where
[TABLE]
Parametrizing z_{1}(\theta_{1})=\frac{1}{I+1}+\big{(}\frac{I}{I+1}-\frac{\beta}{\sqrt{t-s+1}}\big{)}e^{\mathbf{i}\theta_{1}}, we need the following lemma.
Lemma 7.7**.**
There exists positive such that
[TABLE]
for small enough and large enough.
Proof.
Similar to the proof of Lemma 7.4, it suffices to show there exists positive constants such that
[TABLE]
We prove the lemma for , ( small) and ( large) respectively
- •
.
- •
( small): There exists and constants and such that (7.39) holds for .
- •
( large): There exists such that \big{|}\mathfrak{D}_{\epsilon}(z(\theta))\big{|},\big{|}\mathfrak{H}_{\epsilon}(z(\theta))\big{|}<1-\delta for .
We consider the first two bullet points and ( small). The analysis of and ( small) case for is similar to Lemma 7.4, we do not repeat here. For , by straightforward calculation,
[TABLE]
For the first equation above, we taylor expand at and according to (7.44),
[TABLE]
For , taylor expanding around , according to (7.44),
[TABLE]
Combining this with yields
[TABLE]
Using (7.41), (7.42) and (7.40), we get
[TABLE]
Moreover, straightforward calculation gives for (which is a small neighborhood of ). Thereby, by Taylor expansion we find that
[TABLE]
Using (7.43), and , we see that there exists such that for large and small,
[TABLE]
which gives the first bullet point.
For ( small), we readily calculate
[TABLE]
Thus, via Taylor expansion, we find that for ,
[TABLE]
which conclude the second bulletin point.
For ( large), recall z(\theta)=\frac{1}{I+1}+\big{(}\frac{I}{I+1}-\frac{\beta}{\sqrt{t-s+1}}\big{)}e^{\mathbf{i}\theta}, we notice that
[TABLE]
Thanks to Lemma 7.5, there exists such that for large enough and small enough,
[TABLE]
which completes our proof. ∎
For (7.38), we show that the indicator prohibits to be too small.
Lemma 7.8**.**
We can choose large enough such that if \big{|}\mathfrak{p}_{\epsilon}(z(\theta))\big{|}>r^{\prime}_{2} with z(\theta)=\frac{1}{I+1}+\big{(}\frac{I}{I+1}-\frac{\beta}{\sqrt{t-s+1}}\big{)}e^{\mathbf{i}\theta}, then .
Proof.
Note that , it suffices to show that
[TABLE]
Referring to (7.28), we taylor expand around
[TABLE]
We highlight that there is no term in the expansion, which is important for our proof.
We taylor expand at . Using (7.44), and , we find that for large enough and small enough,
[TABLE]
In the last inequality, we used the condition . In addition, it is straightforward to see that \frac{d}{d\theta}|\mathfrak{p}_{\epsilon}(z(\theta))|\big{|}_{\theta=0}=0 and there exists such that \big{|}\frac{d^{2}}{d\theta^{2}}|\mathfrak{p}_{\epsilon}(z(\theta))|\big{|}\leq C^{\prime} for . Consequently, via Taylor expansion, for ,
[TABLE]
Consequently, we have that when ,
[TABLE]
By choosing large enough, we see that . ∎
We are ready to prove Theorem 7.3 for case. As , it is enough to bound respectively and . We begin with (7.37). The proof consists a sequence of bounds on terms appearing in the integrand (7.37). We parametrize by z_{1}(\theta_{1})=\frac{1}{I+1}+\big{(}\frac{I}{I+1}-\frac{\beta}{\sqrt{t-s+1}}\big{)}e^{\mathbf{i}\theta_{1}} and .
(): Show that .
Observe that |z_{1}(\theta_{1})|=\big{|}\frac{1}{I+1}+\big{(}\frac{I}{I+1}-\frac{\beta}{\sqrt{t-s+1}}\big{)}e^{\mathbf{i}\theta_{1}}\big{|} reaches its maximum at , hence
[TABLE]
which gives . By , we deduce .
(, ): Show that .
Clearly, is bounded for and .
(, ): Show that \big{|}\mathfrak{F}_{\epsilon}(z_{1},z_{2})\big{|}\leq C+C\sqrt{t-s+1}(|\theta_{1}|+|\theta_{2}|).
To justify this claim, write
[TABLE]
Let us bound each factor on the RHS of (7.46). Referring to (7.30), we know that . Furthermore, we note that
[TABLE]
which implies |z_{2}-z_{1}|\leq C\big{(}\frac{1}{\sqrt{t-s+1}}+|\theta_{1}|+|\theta_{2}|\big{)}.
In addition, we observe that
[TABLE]
Thus, is uniformly bound over . Incorporating the bound for each factor on the RHS of (7.46) gives the desired bound.
(, ): Show that .
This is proved using the same reasoning for (7.17).
(, ): Show that .
The result directly follows from Lemma 7.7. For , note that either or (depending on the choice of ). Lemma 7.4 implies .
Via change of variable and and incorporating the preceding bounds, we arrive at
[TABLE]
Applying change of variable , we conclude
[TABLE]
We turn to study in (7.38). The proof consists of bounds on terms involved in the integral (7.38). In the following we parametrize z_{1}(\theta_{1})=\frac{1}{I+1}+\big{(}\frac{I}{I+1}-\frac{\beta}{\sqrt{t-s+1}}\big{)}e^{\mathbf{i}\theta_{1}}.
(, ) Show that .
By , we deduce that for
(, ): Show that .
By (, ), we see that for . We are left to show for large and small,
[TABLE]
Recall from (7.14) that when is small enough, all the singularity of belongs to the interval for some . As , it suffices to show that
[TABLE]
To justify this, we parametrize by ,
[TABLE]
Hence, we conclude (7.48).
(, ): Show that .
Referring to (7.31),
[TABLE]
Let us first bound . We know from the discussion in () that . It is straightforward that \big{|}\mathfrak{p}_{\epsilon}(z_{1})^{x_{1}-y_{2}}\mathbf{1}_{\{|\mathfrak{p}_{\epsilon}(z_{1})|>r^{\prime}_{2}\}}\big{|}\leq e^{-\frac{\beta}{\sqrt{t-s+1}}|x_{1}-y_{2}|}, which implies
[TABLE]
In addition, one can compute
[TABLE]
recall . This implies that
[TABLE]
Combining (7.49) and (7.50) yields
[TABLE]
(, ): Show that .
This directly follows from Lemma 7.7.
Consequently, we find that
[TABLE]
where the last inequality is due to Lemma 7.8. Via change of variable , we get
[TABLE]
For the second inequality above, we used the fact Thereby,
[TABLE]
Combining this with the upper bound over (7.47) concludes Theorem 7.3 part (a).
For the gradient, note that applying or to (7.37) and (7.38) will gives an additional in the integrand of and , we bound and perform the change of variable produces an extra factor of . Similarly, applying will produce an additional factor . We bound
[TABLE]
performing change of variable produces an extra factor of . This completes the proof of Theorem 7.3 (b), (c).
7.4. Estimate of , the case
We turn to prove Theorem 7.1 when and . This case is more involved than the previous one. One stumbling block is that we prefer to deform the -contour to be to extract the spatial exponential decay. On the other hand, as depicted in Figure 4, the unit circle does not satisfy the steepest descent condition for . We resolve this issue by first shrinking the -contour to , then for , we re-deform the -contour from to .
We define
[TABLE]
recall is some fix constant which belongs to . Since as , it is clear that for large enough, is admissible. Note that the parametrization of is given by the right part of Figure 5.
We decompose ,
[TABLE]
Let us study in the first place. As we mention at the beginning, when , does not favor the contour to extract spatial decay. We prove in the following that we can re-deform the -contour from to .
Lemma 7.9**.**
For large enough and small enough,
[TABLE]
Proof.
The contours and share a common part . We denote by and . Decompose the contour , , it suffices to prove
[TABLE]
To prove the above equation, we first claim that for small enough and large enough,
[TABLE]
That is to say, the -contour is always , which does not depend on the choice of .
To justify this claim, we need to prove for small enough and large enough
[TABLE]
We denote by , and . Note that as and ,
[TABLE]
Therefore, it suffices to consider the limit case and show that there exists s.t.
[TABLE]
If , we parametrize , where for some positive constant . We readily compute
[TABLE]
If , we parametrize where for some positive constant .
[TABLE]
where the first inequality above is due to the fact that increases as increases.
Having shown (7.53), by Fubini’s theorem, the desired identity (7.52) turns into
[TABLE]
In order to justify the identity above, it is sufficient to show that for all ,
[TABLE]
which is equivalent to
[TABLE]
where is the boundary of the crescent , which is depicted in Figure 6 (note that ).
We set out proving (7.54). Since is a closed curve, according to Cauchy’s theorem, we only need to prove that no pole of the integrand (7.54) lies inside of . As we mentioned before, for small enough, the pole either equals or belongs to . It is straightforward that . Hence, we only need to show that for all .
We claim that for large enough and small enough,
[TABLE]
Note that as and ,
[TABLE]
where and . Therefore, it suffices to show that
[TABLE]
To justify the inequality above, we first observe that . In addition, by setting , we see that
[TABLE]
Consequently, we proved , which completes the proof for Lemma 7.9. ∎
In summary, we can write , where
[TABLE]
and is given by (7.51).
Lemma 7.10**.**
For the parametrization given in Figure 5, we have for large enough and small enough
[TABLE]
Proof.
Similar to Lemma 7.7, it suffices to show that there exists s.t.
[TABLE]
We split out proof for , for ( small) and for ( large).
- •
.
- •
( small): There exists and constants and such that (7.39) holds for .
- •
( large): We can find such that \big{|}\mathfrak{D}_{\epsilon}(z(\theta))\big{|},\big{|}\mathfrak{H}_{\epsilon}(z(\theta))\big{|}<1-\delta for .
Recall that is the same as in a neighborhood of , hence when is small. This being the case, the proof for and ( small) is the same as in Lemma 7.7. For ( large), since when and satisfies the steepest descent condition, we find that for large and small,
[TABLE]
This completes our proof. ∎
We begin to estimate in (7.55). In what follows, we check a sequence of bounds on terms involved in the integral (7.55), we parametrize and .
(): Show that .
Since and , we have . Along with the condition for , we obtain .
(): Show that \big{|}\mathfrak{F}_{\epsilon}(z_{1},z_{2})\big{|}\leq C+C\sqrt{t-s+1}(|\theta_{1}|+|\theta_{2}|).
The argument for this part is the same as in the case.
: Show that .
The argument is the same as case .
(): Show that .
This is the content of Lemma 7.4.
As a consequence, we perform the same procedure as in the case and get
[TABLE]
We turn our attention to study , the proof similarly consists of bounds on terms involved in the integral (7.51). In the following we parametrize as depicted in Figure 5.
(, ): Show that .
This is by the same argument as in the case.
: Show that .
The argument for this part is the same as in the case.
(, ): Show that .
This is the content of Lemma 7.10.
(): Show that .
Similar to the discussion in () for the case, it is sufficient to show
[TABLE]
Since for , could be much less than , we can not bound and separately. Instead, we write
[TABLE]
Note that (since and ), hence
[TABLE]
We claim that
[TABLE]
Once this is proved, by (7.57)
[TABLE]
Let us justify (7.58). We decompose , where and . If , we reparametrize by . It suffices to show that
[TABLE]
By straightforward computation, one sees that reaches its minimum at . Hence we only need to prove that
[TABLE]
By (7.44), . In addition, direct computation yields and uniformly bounded in a small neighborhood of . Consequently, we taylor expand at ,
[TABLE]
for large and small.
If , which means that . We see that
[TABLE]
We claim that for , \big{|}\frac{z_{1}}{z_{1}+I-1}\big{|}>\frac{1}{I}. This could verify by inserting into (7.59). A geometric way to prove this inequality is that one has for all satisfying . If ones increase the radius of circle (by ), the value of \big{|}\frac{z}{z+I-1}\big{|} will also increase. Thereby,
[TABLE]
This implies when , for large and small, which completes the proof of (7.58).
Similar to the proof of Lemma 7.8 in the case, we find that , hence
[TABLE]
Combining the bounds (7.56) and (7.60) implies Theorem 7.3 (a).
To estimate the gradient, the procedure is similar to in case, note that applying or to (7.55) and (7.51) gives an additional factor, applying produces an additional factor . By , we conclude Theorem 7.3 (b), (c).
7.5. Estimate of , the case
In this section, we fix in (7.29). Note that , the difficulty for this case is to choose a suitable -contour so as to extract the spatial decay from in the integrand (7.32). Let us write
[TABLE]
We control respectively and . We deform the -contour to
[TABLE]
where is a positive constant that we will specify later. Note that when , this contour can only be implicitly defined (when it is a circle). The following lemma provides a few properties of the contour.
Lemma 7.11**.**
For large enough and small enough, given , there exists a unique positive such that
[TABLE]
* is infinitely differentiable with . Moreover, one has uniformly for ,*
[TABLE]
where represents the -th derivative of .
Proof.
Let , as and , the equation converges to
[TABLE]
(note and ). Setting in (7.62) yields
[TABLE]
Factorizing the LHS of (7.63) yields
[TABLE]
Thus, (7.63) permits four root at
[TABLE]
We only care about positive root, thus the contour (7.62) can be parametrized by .
Similarly, inserting in (7.61) yields
[TABLE]
where are constants depending on that converge to the coefficient in (7.63):
[TABLE]
Denote by
[TABLE]
By (7.65), when is small and is large, and . By continuity, for each , admits a positive root. Since is a perturbation of , as and , the roots of converge to those in (7.64), which implies the the positive root of is unique for small and large. We denote this unique positive root by . It is also clear that for
[TABLE]
Moreover, for all , is a simple root of . Hence, \frac{\partial}{\partial r}P(\theta,r)\big{|}_{r=\frac{I}{I+1}}\neq 0, using implicit function theorem shows that for small and large, is smooth over . Furthermore,
[TABLE]
In addition, by (7.66) and implicit function theorem, uniformly over
[TABLE]
this completes our proof. ∎
We adopt the parametrization . From the preceding lemma, as and , , thus the contour is admissible for small and large. As before, we decompose , where
[TABLE]
Lemma 7.12**.**
There exists (which depends on ) such that for large enough and small enough, we have
[TABLE]
Proof.
Consider an alternate parametrization , where the existence and uniqueness of are confirmed by Lemma 7.11. It suffices to show for large enough and small enough,
[TABLE]
We prove (7.69) in two steps.
- •
First, .
- •
Second, for .
We verify the first bullet point. Note that uniformly in an neighborhood of ,
[TABLE]
Referring to (7.33), \frac{d}{dz}z\mathfrak{p}_{*}(z)\big{|}_{z=1}=2. Thus, there exists such that for small enough and ,
[TABLE]
We taylor expand around ,
[TABLE]
Referring to (7.44), we see for small enough, which implies
[TABLE]
Hence, . We have by (7.70) and (7.71)
[TABLE]
The first inequality yields
[TABLE]
which gives the lower bound. The second inequality indicates that (by (7.44))
[TABLE]
Owing to , we see that for constant large enough, which concludes the first bullet point.
We move on proving the second bullet point. We set . When small and large, we readily compute (note that is nearly and approximates )
[TABLE]
which implies that reaches its minimum at . In other words, . In addition, . By intermediate value theorem, for each fixed , the equation admits a root . By uniqueness, this root equals , thereby for all . ∎
Lemma 7.13**.**
For large enough, large enough and small enough, the condition with implies .
Proof.
The proof is similar to Lemma 7.8. Since , we have Hence, . It suffices to show that
[TABLE]
Referring to (7.45), we see that
[TABLE]
By (7.44), we see for some positive constant , together with the fact
[TABLE]
we obtain
[TABLE]
In addition, by Lemma 7.11, . Using this, it is straightforward to compute \frac{d}{d\theta}|\mathfrak{p}_{\epsilon}(z(\theta))|\big{|}_{\theta=0}=0 and there exists such that \big{|}\frac{d^{2}}{d\theta^{2}}|\mathfrak{p}_{\epsilon}(z(\theta))|\big{|}\leq C^{\prime} for . Consequently, one has by taylor expansion
[TABLE]
Thereby, we can pick large enough s.t. implies . ∎
Lemma 7.14**.**
For large and small, there exists positive constants such that
[TABLE]
Proof.
Similar to Lemma 7.7, it suffices to show that there exists s.t.
[TABLE]
We split out proof for , for ( small) and for ( large).
- •
.
- •
( small): There exists and constants and such that (7.39) holds for .
- •
( large): There exists such that \big{|}\mathfrak{D}_{\epsilon}(z(\theta))\big{|},\big{|}\mathfrak{H}_{\epsilon}(z(\theta))\big{|}<1-\delta for .
Owing to Lemma 7.12, , hence the argument for is similar to Lemma 7.10.
For ( small), using Lemma 7.11, one has
[TABLE]
Using this, after a tedious but straightforward calculation (recall ),
[TABLE]
The last line holds for all where is a constant. Hereafter, the argument is same as in Lemma 7.7, we do not repeat it here.
For ( large), since
[TABLE]
we have
[TABLE]
By the steepest descent condition (SD), we conclude ( large). ∎
Now we are ready to bound and . We begin with given by (7.67). The proof consists of bounding each terms involved in the integrand (7.67). We parametrize , .
(): Show that .
By Lemma 7.12, we see that , since equals or , we find that , which implies .
(): Show that \big{|}\mathfrak{F}_{\epsilon}(z_{1},z_{2})\big{|}\leq C+C\sqrt{t-s+1}(|\theta_{1}|+|\theta_{2}|).
By the argument in () in () case. It suffices to show that . Note that
[TABLE]
By Lemma 7.11 and Lemma 7.12, we know that and uniformly for , we see that
[TABLE]
Since or , we have
[TABLE]
Incorporating the bound (7.73) and (7.74) into the RHS of (7.72), we conclude .
: Show that .
This is the same as case .
(): Show that .
This is the content of Lemma 7.14.
Consequently, we perform the same procedure as in the case and get
[TABLE]
Let us move on bounding with integral expression (7.68). We parametrize by
(, ): Show that .
This is by the same argument as in the case.
: Show that .
The argument for this is the same as in the case.
(, ):Show that .
This is the content of Lemma 7.14.
(): Show that .
By the discussion in (), It is sufficient to show that . We write
[TABLE]
Since , . In addition, referring to Lemma 7.12, one has . Consequently,
[TABLE]
Thereby, using the same manner as case,
[TABLE]
We conclude Theorem 7.3 (a).
To estimate the gradient, the procedure is similar to in case, note that applying or will give an additional factor , while applying will produce an additional factor . By , we conclude Theorem 7.3 (b), (c).
8. Proof of Proposition 6.8 via self-averaging
In this section, we apply the two Markov dualities in Corollary 3.9 and the estimate of in Theorem 7.1 to conclude Proposition 6.8. The first step is to expand the term and .
8.1. Expanding and
We use to denote a generic uniformly bounded (random) process, which may differ from line to line. Define
[TABLE]
Referring to (5.10) for the expression of
[TABLE]
Here, we used the relation and then changed the order of summation in the last equality.
Likewise, by the expression (5.11) of
[TABLE]
Using Lemma 5.4, one has and . Consequently,
[TABLE]
For and , we denote by
[TABLE]
Lemma 8.1**.**
Recall from (6.22) that
[TABLE]
we have
[TABLE]
Proof.
We name the three terms on the RHS of (8.1) (from left to right) as respectively and those on the RHS of (8.2) as , , . Multiplying (8.1) by (8.2) gives
[TABLE]
Expanding this product, it is straightforward that
[TABLE]
The sum of the rest of terms equals
[TABLE]
Therefore, we find that
[TABLE]
Thus,
[TABLE]
Adding to both sides yields
[TABLE]
We claim that
[TABLE]
If (8.7) holds, note that
[TABLE]
Replacing the term in the LHS of (8.6) by the RHS of (8.7), we prove Lemma 8.1.
To justify (8.7), we write
[TABLE]
Let us analyze the first and second term on the RHS of (8.8) respectively. For the second term, we compute
[TABLE]
Here, we used , the expression of which is given in (5.1). Using the preceding equation, we find that
[TABLE]
Due to Lemma 5.4,
[TABLE]
Thereby, for the second term on the RHS of (8.8),
[TABLE]
For the first term on the RHS of (8.8), noticing (recall ), hence
[TABLE]
Since ,
[TABLE]
Note that for any , there exists a constant such that
[TABLE]
Thus, if , one has
[TABLE]
If , one simply has
[TABLE]
Therefore,
[TABLE]
Referring to (8.9) for the expression of , using (7.4) we see that there exists s.t. for small enough and for all
[TABLE]
Combining this with (8.11) gives
[TABLE]
Combining this with (8.10), we prove the desired claim (8.7). ∎
By Lemma 8.1, we reduce the proof of Proposition 6.8 to the following lemmas.
Lemma 8.2**.**
For any given , there exists positive constants and such that for all ,
[TABLE]
where we used the shorthand notation .
Lemma 8.3**.**
Fix , there exists positive constants and such that for all and ,
[TABLE]
We will prove Lemma 8.2 and Lemma 8.3 in the next two sections. Let us first conclude Proposition 6.8 based on them.
Proof of Proposition 6.8.
Referring to Lemma 8.1, we have
[TABLE]
By Lemma 8.2 and Lemma 8.3, together with the bound \big{\|}Z(s,x^{\star}(s))\big{\|}_{2}\leq Ce^{u\epsilon|x^{\star}|} (which follows from Proposition 6.1), one has
[TABLE]
Using , we obtain
[TABLE]
This completes the proof of Proposition 6.8. ∎
8.2. Proof of Lemma 8.2
Recall the notation , we see that by Taylor expansion
[TABLE]
Hence,
[TABLE]
We will use these elementary relations frequently in the sequel.
The following lemma is crucial for the proof of Lemma 8.2.
Lemma 8.4**.**
Given and , there exists constant and such that for all such that for ,
[TABLE]
For ,
[TABLE]
Proof.
Let us first justify (8.17). Recall the two point duality (5.21),
[TABLE]
As , it is straightforward that by this duality, if ,
[TABLE]
If ,
[TABLE]
We assume without loss of generosity, the proof of (8.17) for will be similar (one only needs to replicate the estimate of to ). By the estimate of provided in Theorem 7.1 (b), we see that
[TABLE]
This, together with the moment bound of in (6.1) yields
[TABLE]
Due to Lemma 6.3, we see that we can choose large enough so that
[TABLE]
Thus,
[TABLE]
Referring to (8.19), we conclude (8.17).
We turn our attention to prove (8.18). With the aid of (5.21), one has for ,
[TABLE]
Note that (8.20) does not hold when (see Remark 8.5 below). Theorem 7.1 implies
[TABLE]
By same argument used in proving (8.17), one has
[TABLE]
This concludes the proof of the lemma. ∎
With the help of the preceding lemma, we proceed to prove Lemma 8.2.
Proof of Lemma 8.2.
Referring to (8.3), (8.4) that
[TABLE]
By triangle inequality, one has
[TABLE]
To prove Lemma 8.2, it is sufficient to show that there exists constant such that for all , and some constant ,
[TABLE]
Note that here, we include in (8.21), which is not needed to prove Lemma 8.2. We are going to use this in the proof of Lemma 8.3.
We begin with proving (8.21), by writing
[TABLE]
Using (8.12) to bound , one has
[TABLE]
Let us analyze the two terms on the RHS of (8.23) respectively. For the first term, via Cauchy-Schwarz inequality |\mathbb{E}\big{[}XY\big{]}|\leq\big{\|}X\big{\|}_{2}\big{\|}Y\big{\|}_{2}, one has
[TABLE]
By the moment bound in Proposition 6.1, we have \big{\|}Z_{\nabla}(s,x_{1},x_{2})\big{\|}_{2}\leq Ce^{u\epsilon(|x_{1}|+|x_{2}|)}. Combining this with (8.17),
[TABLE]
Consequently, the first term in (8.23) is upper bounded by
[TABLE]
where in the second inequality above we used the integral approximation
[TABLE]
and in the last inequality we used .
Using again \big{\|}Z_{\nabla}(s,x_{1},x_{2})\big{\|}_{2}\leq Ce^{u\epsilon(|x_{1}|+|x_{2}|)}, the second term in (8.23) is readily upper bounded by
[TABLE]
Incorporating the bounds (8.24) and (8.25) into the RHS of (8.23), we get (8.21).
We proceed to justify (8.22), the method is similar to the proof of (8.21). Write
[TABLE]
Using again (8.12), one has
[TABLE]
Let us analyze the two terms on the RHS of (8.26) respectively. For the first term, by Cauchy Schwarz,
[TABLE]
Using the bound \big{\|}Z_{\nabla}(s,x_{1},x_{2})\big{\|}_{2}\leq Ce^{u\epsilon(|x_{1}|+|x_{2}|)} and (8.18), we have
[TABLE]
Therefore,
[TABLE]
In the second inequality above, we used the integral approximation
[TABLE]
For the second term in (8.26), it is clear that
[TABLE]
Incorporating the bounds (8.27) and (8.28) into the RHS of (8.26), we prove the desired (8.22). ∎
Remark 8.5**.**
In the argument above, we showed vanishes after averaging over a long time interval when . The readers might wonder whether the same holds for ? The answer is negative. In the case , we used two particle duality (5.21) to move the gradient from to
[TABLE]
However, if , the same two particle duality gives instead
[TABLE]
The same argument fails because we do not have an effective estimate of
[TABLE]
In fact, when , does not vanish after averaging. One quick way to see this is to use
[TABLE]
where represents the fractional part of . This implies that is lower bounded by a constant times , which does not vanish after averaging.
8.3. Proof of Lemma 8.3
The aim of this section is to justify Lemma 8.3, which indicates that vanishes after averaging over a long time interval. This was proved for the stochastic six vertex model [CGST18] (which corresponds to ). Note that when , for all one has , which yields . [CGST18] utilizes this crucial observation to show that
[TABLE]
where in the last line above, we used (8.15). We have seen in the previous section that vanishes after averaging, which implies that will also vanish.
When , can takes more than two values so the relation no longer holds. Notice that in the proof of Lemma 8.2, we have only leveraged the first duality (5.21) in the Lemma 5.2. To conclude Lemma 8.3, we will combine both of the dualities (5.21) and (5.22).
Before moving to the proof, we first offer a heuristic argument to explain why the is the value which makes vanish after averaging.
Heuristic argument for Lemma 8.3.
Note that
[TABLE]
In Theorem A.3, we find that the stationary distribution of the (bi-infinite) SHS6V model is given by , where is defined in (A.1). It is straightforward to verify that is near stationary with density (Definition 5.5). Start the SHS6V model from , by stationarity for all and . Heuristically, we can approximate by \mathbb{E}_{\pi_{\rho}}\big{[}(\widetilde{\eta}_{x+1}(t)-\rho)^{2}\big{]}Z(t,x)^{2}. Note that
[TABLE]
where \text{Var}\big{[}\pi_{\rho}\big{]} represents the variance of the probability distribution . Referring to Lemma A.2, we have
[TABLE]
where is the unique negative real number satisfying It is straightforward that under weak asymmetric scaling (5.30), one has . Therefore,
[TABLE]
which explains . ∎
We proceed to prove Lemma 8.3 rigorously. The first step is to express in terms of the two duality functionals in Lemma 5.2,
[TABLE]
In the last equality, we replaced by , according to (8.16), this procedure produces an error term which can be absorbed in the .
Recall that . Under weak asymmetric scaling, , one has
[TABLE]
These approximations imply that
[TABLE]
where we recall the expression of the functional from (5.19). Inserting (8.31) into the RHS of (8.29)
[TABLE]
Recall that our goal is to show
[TABLE]
Referring to the expression of in (8.5), we need to prove that there exists some such that for all ,
[TABLE]
Using (8.32), it suffices to show that for all ,
[TABLE]
and
[TABLE]
Note that (8.34) is proved by taking in (8.21). Therefore, we only need to prove (8.33). Similar to the proof in Lemma 8.2, to conclude (8.33), it suffices to prove the following lemma for upper bounding the conditional expectation. We do not repeat the rest of the proof here.
Lemma 8.6**.**
For and , there exists constant and such that for all and ,
[TABLE]
Proof.
Combining both of the dualities (5.21) and (5.22), one has
[TABLE]
We split the summation above according to the range of the value of ,
[TABLE]
We name the terms on the RHS of (8.36) and respectively and we bound them separately. It follows from Proposition 6.1 that
[TABLE]
Invoking Theorem 7.1 (a) and Lemma 6.3, we find that
[TABLE]
We proceed to bound , recall that when ,
[TABLE]
which could be rewritten as (using (8.30))
[TABLE]
Consequently, we write
[TABLE]
It Is straightforward by (8.15) and (8.16) that
[TABLE]
Inserting these into the RHS of (8.38),
[TABLE]
Let us name respectively the three terms on the RHS above to be , , . Recall the summation by part formula (with notation )
[TABLE]
Note that
[TABLE]
by (8.39), we move the gradient from to ,
[TABLE]
Using Theorem 7.1 part (a) and part (b) to control and respectively, we see that for ,
[TABLE]
Applying Lemma 6.3 yields \big{\|}\mathbf{J_{1}}\big{\|}_{n}\leq\frac{C\epsilon^{-\frac{1}{2}}}{\sqrt{t-s+1}}e^{2u\epsilon|x|}. Likewise, we obtain \big{\|}\mathbf{J_{2}}\big{\|}_{n}\leq\frac{C\epsilon^{-\frac{1}{2}}}{\sqrt{t-s+1}}e^{2u\epsilon|x|}.
For , applying Theorem 7.1 part (a) and Lemma 6.3 implies that
[TABLE]
In the last inequality above, we used the fact , which implies .
Combining the bounds for \big{\|}\mathbf{J_{1}}\big{\|}_{n}, \big{\|}\mathbf{J_{2}}\big{\|}_{n}, \big{\|}\mathbf{J_{3}}\big{\|}_{n}, we have
[TABLE]
Recall from (8.36) that
[TABLE]
combining the bounds for and in (8.40) and (8.37), we conclude the desired (8.35). ∎
Appendix A Stationary distribution of the SHS6V model
In this section, we provide a one parameter family of stationary distribution for the unfused SHS6V model. It is worth to remark that in the recent work of [IMS19], a translation-invariant Gibbs measure was obtained (using the idea from [Agg18a]) for the space-time inhomogeneous SHS6V model on the full lattice, see Proposition 4.5 of [IMS19]. However, It is not obvious that the dynamic of SHS6V model under this Gibbs measure coincides with the one of the bi-infinite SHS6V model specified in Lemma 2.1. This being the case, we choose to proceed without relying on the result from [IMS19].
We start with a well-known combinatoric lemma.
Lemma A.1** (q-binomial formula).**
Set as usual, the following identity holds for all ,
[TABLE]
Proof.
According to -binomial theorem [AAR00],
[TABLE]
When , for . Therefore,
[TABLE]
∎
Lemma A.2**.**
Fix , and , define a probability measure on :
[TABLE]
where is the unique negative real number satisfying
[TABLE]
Furthermore, we have
[TABLE]
Proof.
We first show that is indeed a probability measure. It is clear that for all . By Lemma A.1,
[TABLE]
Next, we compute the expectation and the variance of . Using again Lemma A.1, the moment generating function is given by
[TABLE]
It is clear that
[TABLE]
Via (A.3), one has
[TABLE]
Note that
[TABLE]
combining this with (A.2) yields
[TABLE]
which concludes the lemma. ∎
Theorem A.3**.**
For , the product measure is stationary for the unfused SHS6V model (Definition 2.3).
Proof.
It suffices to show that if , then .
Recall that records the number of particles (either zero or one) that move across location at time . We first show that (recall that ). To this end, referring to (2.4),
[TABLE]
Recalling from (2.1), B(t,z,\eta)\sim\text{Ber\,}\big{(}\frac{\alpha(t)(1-q^{\eta})}{1+\alpha(t)}\big{)},B^{\prime}(t,z,\eta)\sim\text{Ber\,}\big{(}\frac{\alpha(t)+\nu q^{\eta}}{1+\alpha(t)}\big{)}. Since the random variables are all independent,
[TABLE]
Therefore, by tower property
[TABLE]
As , we obtain using Lemma A.1
[TABLE]
Inserting the value of \mathbb{E}\big{[}q^{\eta_{y}(t)}\big{]} into the RHS of (A.5) yields that
[TABLE]
Since , we conclude that
[TABLE]
The next step is to show that the marginal of is distributed as for each coordinate. Referring to (A.4), it is straightforward that the following recursion holds
[TABLE]
Therefore,
[TABLE]
For the second equality above, we used . Therefore,
[TABLE]
Due to (A.4), we see that K(t,y-1)\in\sigma\Big{(}B(t,z,\eta),B^{\prime}(t,z,\eta),\eta_{z}(t):z\leq y-1,\eta\in\{0,1,\dots,I\}\Big{)}. Note that we have assumed , which implies the independence between and for . Therefore, and are independent. Using (A.8) we get
[TABLE]
By and , one readily has
[TABLE]
To conclude Theorem A.3, it suffices to show the independence among for different value of . It is enough to show that
[TABLE]
We need the following lemma.
Lemma A.4**.**
For all , is independent with .
Let us first see how this lemma leads to (A.9). We have via (A.4),
[TABLE]
Combining this with (A.8),
[TABLE]
Since are all independent for different , one has
[TABLE]
We achieve
[TABLE]
Using Lemma A.4, we conclude
[TABLE]
Therefore,
[TABLE]
On the other hand, by (A.7) and (A.8), we conclude for all
[TABLE]
Combining (A.10) and (A.11), we find that for all
[TABLE]
which concludes (A.9). ∎
Proof of Lemma A.4.
As , it suffices to show that for all , one has
[TABLE]
Due to (A.7),
[TABLE]
Together with (A.8), we obtain that if ,
[TABLE]
If ,
[TABLE]
The discussion above yields (using the independence between and )
[TABLE]
which concludes Lemma A.4. ∎
Remark A.5**.**
Since , it is clear that for all , is also stationary for the fused SHS6V model .
Appendix B KPZ scaling theory
The KPZ scaling theory has been developed in a landmark contribution by [KMHH92]. The scaling theory is a physics approach which makes prediction for the non-universal coefficients of the KPZ equation. In this appendix, we show how the coefficients of the KPZ equation (1.11) arise from the microscopic observables of the fused SHS6V model using the KPZ scaling theory.
Recall that Theorem 1.6 reads
[TABLE]
Here, is the fused height function and solves the KPZ equation
[TABLE]
where
[TABLE]
The first step in the KPZ scaling theory is to derive the stationary distribution of the fused SHS6V model, which is exactly what we did in Appendix A (see Remark A.5). Under stationary distribution , we proceed to define two natural quantities of the models:
- •
The average steady state current is defined as
[TABLE]
where means that we are taking the expectation under stationary distribution and is given in (1.9). Note that under stationary distribution, the average steady state current depends neither on space or time. Let us explain the meaning of (B.1). Note that records the number of particles in the fused SHS6V model that move across location at time , we subtract here because we are in a reference frame that moves to right with speed .
- •
The integrated covariance is defined as
[TABLE]
The KPZ scaling theory (equation (12) and (15) of [KMHH92]) predicts that
[TABLE]
and depend on under weakly asymmetry scaling (5.30).
Let us first verify , note that under stationary distribution, is the sum of i.i.d. random variables with the same distribution , hence A_{\epsilon}(\rho)=\text{Var}\big{[}\pi_{\rho}\big{]}. By Lemma A.2, we know that
[TABLE]
where is the unique negative solution of
[TABLE]
Under weakly asymmetric scaling, one has , which yields . Therefore,
[TABLE]
This matches with the value of .
We proceed to verify . First, note that by ,
[TABLE]
where . We have shown in (A.6) that , where . Therefore,
[TABLE]
which yields
[TABLE]
We proceed to taylor expand around . Note that is implicitly defined through (B.2), we expand around
[TABLE]
Note that depends on through . Via straightforward calculation, one has
[TABLE]
which implies
[TABLE]
Referring to the expression of in (1.9), one has the asymptotic expansion
[TABLE]
Consequently,
[TABLE]
We have
[TABLE]
which coincides with the value of .
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- 1[AAR 00] George E Andrews, Richard Askey, and Ranjan Roy. Special functions , volume 71. Cambridge university press, 2000.
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