Crossover between various initial conditions in KPZ growth: flat to stationary

TL;DR

This paper conjectures a universal probability distribution for the KPZ height function at large times, capturing the crossover between different initial conditions including flat, stationary, and droplet, using a replica Bethe ansatz approach.

Contribution

It introduces a new conjecture for the universal crossover distribution in KPZ growth between flat and stationary initial conditions, derived via a replica Bethe ansatz method.

Findings

Derived the crossover distribution between flat and stationary initial conditions.

Confirmed some cases match known results, validating the approach.

Provided new results for the KPZ universality class in crossover regimes.

Abstract

We conjecture the universal probability distribution at large time for the one-point height in the 1D Kardar-Parisi-Zhang (KPZ) stochastic growth universality class, with initial conditions interpolating from any one of the three main classes (droplet, flat, stationary) on the left, to another on the right, allowing for drifts and also for a step near the origin. The result is obtained from a replica Bethe ansatz calculation starting from the KPZ continuum equation, together with a "decoupling assumption" in the large time limit. Some cases are checked to be equivalent to previously known results from other models in the same class, which provides a test of the method, others appear to be new. In particular we obtain the crossover distribution between flat and stationary initial conditions (crossover from Airy$_1$ to Airy$_{{\rm stat}}$) in a simple compact form.

Equations

$${\partial }_{t}h(x,t)=\nu {\partial }_x^{2}h(x,t)+\frac{{\lambda }_0}{2}({\partial }_{x}h(x,t){)}^2+\sqrt{D}\xi (x,t)$$

$$h(x,t=0)=(h_{0,\mathrm{left}}(x)+\Delta )\theta (-x)+t_0^{1/3}f(x/t_0^{2/3})+(h_{0,\mathrm{right}}(x)-\Delta )\theta (x)$$

$$x^{\ast }=\frac{(2\nu {)}^3}{D{\lambda }_0^2},t^{\ast }=\frac{2(2\nu {)}^5}{D^2{\lambda }_0^4},h^{\ast }=\frac{2\nu }{{\lambda }_0}$$

$${\partial }_{t}h(x,t)={\partial }_x^{2}h(x,t)+({\partial }_{x}h(x,t){)}^2+\sqrt{2}\eta (x,t)$$

$$e^{h(x,t)}=Z(x,t):=\int dy{Z}_{\eta }(x,t|y,0)e^{h(y,t=0)}.$$

$$Z_{\eta }(x,t|y,0)={\int }_{x(0)=y}^{x(t)=x}Dx{e}^{-{\int }_0^{t}d\tau [\frac{1}{4}(\frac{dx}{d\tau }{)}^2-\sqrt{2}\eta (x(\tau ),\tau )]}$$

$${\partial }_{t}Z={\nabla }^{2}Z+\sqrt{2}\eta Z$$

$$h_{\eta }(x,t|y,0)=\ln Z_{\eta }(x,t|y,0)$$

$$h(y,t=0)=h_0(y)=(w_{L}y+a_L{B}_L(-y))\theta (-y)+(-w_{R}y+a_R{B}_R(y))\theta (y)$$

$$h_0^{\Delta }(y)=h_0(y)-\Delta \mathrm{sgn}(y)$$

$${\hat{w}}_{L,R}=t^{1/3}{w}_{L,R},\hat{x}=\frac{x}{2t^{2/3}}$$

$$h(x,t)\simeq v_{\infty }t+O(t^{1/3})$$

$$h(x,t)=-\frac{t}{12}+\tilde{h}(x,t),Z(x,t)=e^{-t/12}\tilde{Z}(x,t)$$

$$h_{drop}(0,t)=\ln Z_{drop}(0,t)\simeq t^{1/3}{\chi }_2+o(t^{1/3}),\mathrm{Prob}({\chi }_2<\sigma )=F_2(\sigma )$$

$$F_2(\sigma )=\mathrm{Det}[I-P_{\sigma }{K}_{Ai}{P}_{\sigma }],K_{Ai}(v,v^{\prime })={\int }_{y>0}dyAi(y+v)Ai(y+v^{\prime })=\frac{Ai(v)A{i}^{\prime }(v^{\prime })-A{i}^{\prime }(v)Ai(v^{\prime })}{v-v^{\prime }}$$

$$h(x,t)\simeq t^{1/3}({\mathcal{A}}_2(\hat{x})-{\hat{x}}^2)+o(t^{1/3})$$

$$h(x,t|y,0)\simeq t^{1/3}({\mathcal{A}}_2(\hat{x}-\hat{y})-(\hat{x}-\hat{y}{)}^2)+o(t^{1/3})$$

$$h_{flat}(0,t)=\ln Z_{flat}(0,t)=2^{-2/3}{t}^{1/3}{\chi }_1+o(t^{1/3}),\mathrm{Prob}({\chi }_1<s)=F_1(s)$$

$$F_1(s)=\mathrm{Det}[1-P_{s/2}{K}^{GOE}{P}_{s/2}],K^{GOE}(v,v^{\prime })=Ai(v+v^{\prime })$$

$$h(x,t)\simeq 2^{1/3}{t}^{1/3}{\mathcal{A}}_1(2^{-2/3}\hat{x})+o(t^{1/3})$$

$$h_{flat}(x,t)=\ln \int dy{e}^{h(x,t|y,0)}\equiv \ln \int dy{e}^{h(y,t|x,0)}\simeq t^{1/3}\underset{\hat{y}}{\max }({\mathcal{A}}_2(\hat{y}-\hat{x})-(\hat{y}-\hat{x}{)}^2)$$

$$\underset{\hat{y}}{\max }({\mathcal{A}}_2(\hat{y})-{\hat{y}}^2)=2^{-2/3}{\chi }_1=2^{1/3}{\mathcal{A}}_1(0)$$

$$Prob(t^{-1/3}(h(x,t)+\frac{x^2}{4t})<\sigma )=\mathrm{Det}[I-P_{\sigma }K{P}_{\sigma }]$$

$${\mathcal{B}}_w(v)=e^{w^3/3}{e}^{-vw}-{\int }_0^{\infty }dyAi(v+y)e^{wy}={\int }_{-\infty }^{0}dyAi(v+y)e^{wy}$$

$$K(v_i,v_j)$$

$$+$$

$$K(v_i,v_j)$$

$$+$$

$$K(v_i,v_j)={\int }_0^{+\infty }dyAi(v_i+y)Ai(v_j+y)-\frac{Ai(v_i)Ai(v_j)}{{\hat{w}}_L+{\hat{w}}_R}+Ai(v_i){\mathcal{B}}_{{\hat{w}}_L+\hat{x}}(v_j)+{\mathcal{B}}_{{\hat{w}}_R-\hat{x}}(v_i)Ai(v_j)$$

$$\hat{\Delta }=\frac{\Delta }{t^{1/3}}$$

$$K(v_i,v_j)$$

Full text

I Introduction

The one-dimensional Kardar-Parisi-Zhang (KPZ) equation KPZ describes, in the continuum, the stochastic growth of an interface, of height $h(x,t)$ at point $x\in\mathbb{R}$, as a function of time $t$

[TABLE]

driven by a unit white noise $\overline{\xi(x,t)\xi(x^{\prime},t^{\prime})}=\delta(x-x^{\prime})\delta(t-t^{\prime})$. It has a number of experimental realizations exp4 ; exp5 ; KPZCoffee ; KPZSeverine and is at the center of large (and growing) universality class, which contains exactly solvable models in discrete settings, studied both in physics and mathematics. Recently there was progress in finding exact solutions for the continuum KPZ equation itself. While the scaling exponents $h\sim t^{1/3}$, $x\sim t^{2/3}$ have been known for a while exponent , the present interest is to characterize the full statistics of the height field $h(x,t)$. The KPZ equation can be mapped to the continuous directed polymer (DP) in a quenched random potential, such that $h(x,t)=\ln Z(x,t)$ is proportional to the free energy of the DP of length $t$ with one fixed endpoint at $x$.

Interestingly, the KPZ interface retains some memory of the initial condition, and a few main universal statistics are found to emerge at large time, depending on the type of initial conditions. Remarkably, these are also related to the universality of large random matrices. This was first obtained from discrete models in the KPZ universality class, i.e. expected to share all its (rescaled) large time properties, such as the PNG growth model png ; spohn2000 ; ferrari1 , the TASEP particle transport model spohnTASEP ; ferrariAiry ; Airy1TASEP or discrete DP models Johansson2000 ; spohn2000 . Recently it was obtained more directly, from exact solutions of the KPZ equation: on the infinite line there are three main classes

• •

The droplet (or hard wedge) initial condition (DP with two fixed endpoints) leads to height fluctuations governed at large time by the Tracy Widom (TW) distribution $F_{2}$, the CDF (cumulative distribution function) of the largest eigenvalue of the GUE random matrix ensemble TW1994Airy . It was solved by two methods (i) as a limit from an ASEP model with weak asymetry spohnKPZEdge leading to a rigorous derivation corwinDP ; reviewCorwin (ii) using the replica Bethe ansatz (RBA) method we ; dotsenko by calculating the integer moments of $Z=e^{h}$ from the known exact solution of the Lieb-Liniger delta Bose gas ll (also derived recently from the sine-Gordon field theory sineG ). Both methods obtained the CDF for all times $t$, as a Fredholm determinant, displaying convergence to $F_{2}$ as $t\to+\infty$.

• •

The flat initial condition (point to line DP), solved with the RBA at all times we-flat ; we-flatlong ; flatnumerics ; cl-14 and at large time dotsenkoGOE . Rigorous calculations within ASEP have not yet led to a proof of the finite time results for the KPZ equation (see Quastelflat for the present status of rigorous calculations). The convergence of the one-point CDF is now to $F_{1}$, associated with the GOE ensemble of random matrices.

• •

The stationary i.e. Brownian initial condition solved first at all times using the RBA SasamotoStationary exhibits convergence at large time to the Baik-Rains $F_{0}$ distribution. Recently it was solved rigorously as a limit of discrete directed polymer models using tools from the theory of Macdonald processes BCFV .

The RBA also allowed to solve the KPZ equation on the half-line we-halfspace which relates to the GSE random matrix ensemble TW1996GOEGSE .

Although non-rigorous (since the integer moments $\overline{Z^{n}}$ of the continuum KPZ equation grow too fast with $n$, as $\sim e^{cn^{3}}$, to determine uniquely the distribution) the RBA has shown impressive heuristic value, often preceding rigorous results, still not available in all cases. A number of the latter have been obtained recently as limits (e.g. $q\to 1$) from a hierarchy of (often novel) integrable discrete models (including $q$-TASEP, $q$-bosons, semi-discrete DP, vertex models) and new mathematical tools (e.g. Macdonald processes) BorodinMacdo ; BorodinQboson ; Duality ; povolo ; BorodinQHahnBethe ; Borodin6Vertex ; CorwinVertexModels .

Besides the three main classes, one expects universal crossover classes (also called transition classes) with initial conditions interpolating from one of the three classes at $x=-\infty$ to another one at $x=+\infty$, see e.g. Fig. 4 in Ref. reviewCorwin . If the two limits are distinct classes, there are three possibilities as follows:

• •

Droplet to stationary: The KPZ equation with half-Brownian initial conditions was solved for all times using the RBA SasamotoHalfBrownReplica . Although the precise form of the obtained kernel is different, it is found equivalent to the result obtained by taking the weak asymmetry limit QuastelAi2BM of the general result for the ASEP with half-Bernouilli initial conditions obtained in TW-ASEPHalfBrownian . At large time this leads to the universal GUE to stationary crossover distribution. It admits an interesting generalization where the initial condition on the half-space is the partition sum of an O’Connel-Yor directed polymer with $N$ layers, equivalently the highest eigenvalue of the $GUE(N)$ Dyson Brownian motion ($N=1$ corresponds to the half-Brownian) BCF .

• •

Droplet to flat: We studied recently using RBA PLDCrossoverDropFlat the transition from GUE to GOE in the KPZ equation, realized for the so-called ”half-flat” intial condition, which is flat for $x<0$ and droplet-like for $x>0$. From the ”half-flat” formula obtained in we-flatlong , we could produce a conjecture for the PDF in the large time limit. We obtained a new formula for the transition Kernel and showed that it is equivalent to the one obtained in Ref. BorodinAiry2to1 from a solution of the TASEP with initial condition of particles on even sites for $x\leq 0$ and empty for $x>0$. This is a mark of the expected universality at large time of this transition class. The corresponding Airy process was defined and characterized in Ref. BorodinAiry2to1 and called ${\cal A}_{2\to 1}$.

• •

Flat to stationary: At present there is no derivation of the flat to stationary distribution directly for the KPZ equation. In the large time limit, the corresponding distribution was obtained in Borodin2speed for TASEP with initial conditions $2\mathbb{Z}_{-}$, i.e. particles on even sites for $x\leq 0$, no particles for $x>0$, with the $M$ first particles endowed with a slower speed $\alpha<1$. With this setting there is a point in the phase diagram of the model where the crossover flat to stationary can be attained (the corresponding kernel is given by Eq. (5.4)-(5.8) there with $M=1$ and $\kappa=0$, for $\alpha=1/2$). In terms of process this is called ${\cal A}_{1\to{\rm stat}}$.

The aim of this paper is to revisit these crossover classes starting from the KPZ equation (1) and using the replica Bethe ansatz method. We will in particular obtain the lacking result for the third universal crossover class in the KPZ equation, the flat to stationary. Note that for each class of initial condition there are two degrees of freedom which can be varied, corresponding to two known invariances of the KPZ equation, namely (in units such that $\lambda_{0}=D=2$, $\nu=1$ - see below) a shift in height by a constant $h(x,t)\to h(x,t)+\Delta$, and a tilt by a finite slope $h(x,t)\to h(x+2wt,t)+wx+w^{2}t$, also known as Galilean invariance on the associated Burgers equation (the derivative of the KPZ equation). Hence it is quite natural to study the crossover between initial conditions with different slopes $w_{L,R}$ and a height mismatch (step size) $2\Delta$ on each side. By scaling the slopes and step size appropriately with time one obtains universal crossover distributions at large time (it then makes sense as a crossover distribution even when the class is the same on each side). We also obtain for example the universal distribution for the wedge initial condition (flat to flat crossover) and for the Brownian to Brownian. In summary we expect our results to apply to any initial condition $h(x,t=0)$ which interpolates between a left initial condition $h_{0,\rm left}(x)$ for $x<0$, and a right initial condition $h_{0,\rm right}(x)$, for $x>0$, each belonging to one the three main classes, with possibly a mismatch in height, $2\Delta$, and in slopes $w_{L}-w_{R}$, merging within an interpolation region of size $x_{0}=t_{0}^{2/3}$, e.g. an initial condition of the form

[TABLE]

where $f(x)$ is a bounded function which decays to [math] at infinity, and $\theta(x)$ the Heaviside unit step function. We will consider the large time limit $t\gg t_{0}$, in which the precise form $f(x)$ of the interpolating region becomes irrelevant, and study the result as a scaling function of the scaled parameters $\Delta/t^{1/3}$ and $w_{L,R}t^{1/3}$.

We obtain these results within the RBA using an assumption in the large time limit, sometimes called a ”decoupling assumption” dotsenkoGOE ; ps-2point ; ps-npoint ; dotsenko2pt ; Spohn2ptnew ; dotsenkoEndpoint ; KPZFixedPoint . The method is similar from the one we applied to study the droplet to flat crossover in Ref. PLDCrossoverDropFlat (some results of that work are recovered here in some limit) but is significantly more involved as it requires working with the combinatorics of groups of replica, as pionneered by Dotsenko dotsenkoGOE (an approach that we will test and slightly extend here). For the Brownian to Brownian crossover, our results take a different form, but agree with known results, e.g. the one of SasamotoStationary , which provides yet another a non-trivial check of the method.

The outline of the paper is as follows. In section II we recall the model, the units and the mapping to the directed polymer. In section II.2 we describe the initial conditions studied in this paper. In section II.3 we first recall standard results, then we summarize the main results of the present work. In section II.4 we give the definitions of the generating functions. Section III contains the main replica calculations. The quantum mechanical method and the Bethe ansatz are recalled, then in section III.2 we display the combinatorics identity which is used, leading to a general formula for the moments and generating function in III.4. The large time limit is studied in section III.5 and using the decoupling assumption leads to an expression for the generating function as a Fredholm determinant, involving a kernel which is given in two equivalent forms. The section IV details these two equivalent forms of the kernel in each of the various cases, with their limit forms and comparison with known results. Finally, the section V is the conclusion. The first Appendix details the combinatorial identity. The next two appendices detail the calculations of the kernels associated to the standard generating function, and the last one for the generalized generating function.

II Model and main results

II.1 KPZ equation, directed polymer and units

Let us consider the KPZ equation (1), and define the scales

[TABLE]

which we will use as units, i.e. we set $x\to x^{*}x$, $t\to t^{*}t$ and $h\to h^{*}h$ so that from now on $x,t,h$ are in dimensionless units and the KPZ equation becomes:

[TABLE]

where $\eta$ is also a unit white noise $\overline{\eta(x,t)\eta(x^{\prime},t^{\prime})}=\delta(x-x^{\prime})\delta(t-t^{\prime})$. As is well known the Cole-Hopf mapping solves the KPZ equation from an arbitrary initial condition as follows. The solution at time $t$ can be written as:

[TABLE]

where here and below we denote $\int dy=\int_{-\infty}^{+\infty}dy$. Here $Z_{\eta}(x,t|y,0)$ is the partition function of the continuum directed polymer in the random potential $-\sqrt{2}~{}\eta(x,t)$ with fixed endpoints at $(x,t)$ and $(y,0)$:

[TABLE]

which is the solution of the (multiplicative) stochastic heat equation (SHE):

[TABLE]

with Ito convention and initial condition $Z_{\eta}(x=0,t|y,0)=\delta(x-y)$. Equivalently, $Z(x,t)$ is the solution of (7) with initial conditions $Z(x,t=0)=e^{h(x,t=0)}$. We will adopt the notation (for the solution of the droplet initial condition started in $y$):

[TABLE]

although it is somewhat improper (it requires a regularization, see below). We will sometimes omit the ”environment” index $\eta$. Here and below overbars denote averages over the white noise $\eta$.

II.2 Initial conditions

We will study the KPZ equation (4) with the following initial condition:

[TABLE]

where $\theta(y)$ is the unit step Heaviside function, $B_{L}(y)$ and $B_{R}(y)$ are independent one-sided unit centered Brownians, with $B_{L}(0)=B_{R}(0)=0$, $B_{L,R}(y)$ being defined for $y\geq 0$. The correlator is $<B_{R}(y)B_{R}(y^{\prime})>=\min(y,y^{\prime})$ and similarly for $B_{L}(y)$. The parameters $w_{L},w_{R}$ (usually chosen positive) measure the bias of the Brownian, i.e. the slopes of the KPZ initial profile on each side.

The parameters $a_{L},a_{R}$ are chosen in $\{0,1\}^{2}$ to allow to study the four ”solvable” cases (in fact three distinct ones, by symmetry). The wedge initial condition corresponds to $a_{L}=a_{R}=0$ and contains left (resp. right) half-flat initial condition as limits $w_{R}\to+\infty$ (resp. $w_{L}\to+\infty$). The Brownian to Brownian (with drifts) corresponds to $a_{L}=a_{R}=1$, and contains the stationary case as a limit (when $w_{L}=w_{R}=0$). The flat to Brownian (with drifts) corresponds to $a_{L}=0,a_{R}=1$ and contains as limits the half-Brownian $w_{L}\to+\infty$ and half-flat $w_{R}\to+\infty$. By symmetry it is also realized for $a_{L}=1,a_{R}=0$.

In addition, at little further expense in the calculation, we will be able to add a step $\Delta$ in the initial height, i.e. study the initial condition

[TABLE]

where $h_{0}(y)$ is any of the above cases. With no loss of generality we will consider $\Delta\geq 0$, i.e. a downward step.

II.3 Results

In order to obtain the most interesting large time limit, we need to scale the original slopes and position with time so that the following rescaled parameters (denoted by hat)

[TABLE]

are kept fixed as time becomes large. This is consistent with the standard KPZ scaling. Clearly this also contains the (less interesting) case where the limit is done instead with fixed $x,w_{L},w_{R}$, which is equivalent to set $\hat{w}_{L,R}\to\infty$ and $\hat{x}\to 0$ in all formula below.

At large time the KPZ field grows linearly in time plus $\sim t^{1/3}$ fluctuations

[TABLE]

and for the continuum KPZ solution 111Note that $v_{\infty}=-1/12$ at large time is a result of the Ito convention in (7) which implies that $\overline{Z(x,t)}$ obeys the free diffusion equation: this defines the Cole-Hopf solution to the continuum KPZ equation for white noise. In presence of a regularized noise (i.e. with spatial correlations) $v_{\infty}$ becomes non-universal. However, as detailed in flatnumerics if one considers $\ln(Z(t)/{\overline{Z}(t)})$ then $v_{\infty}=-1/12$ in our units, $v_{\infty}=-1/12$. To get rid of this part linear in time we will, from now on redefine the KPZ field, and the DP partition sum, at all times, as

[TABLE]

and for notational simplicity, we will omit the tilde in what follow.

II.3.1 Recall of standard results

Let us first recall the standard results. The first is for $h_{drop}(x,t)$ corresponding to the ”droplet” or wedge initial condition $h_{0}(y)=-w|y|$ (i.e. here to $w_{L,R}=w$, $a_{L,R}=0$). Strictly speaking, its exact solution at all times spohnKPZEdge ; corwinDP ; reviewCorwin ; we ; dotsenko is valid only for the ”hard” wedge limit, i.e. $w\to+\infty$. However here we will be interested only in the large time limit, hence $w$ can be chosen arbitrary but fixed. At large time the one-point fluctuations of the height are governed by the GUE Tracy Widom (cumulative) distribution $F_{2}(s)$ as

[TABLE]

where $F_{2}(\sigma)$ is given by a Fredholm determinant involving the Airy Kernel:

[TABLE]

and $P_{\sigma}(v)=\theta(v-\sigma)$ is the projector on $[\sigma,+\infty[$. Note that the solution (8) is $h(x,t|y,0)\equiv h_{drop}(x-y,t)+\ln(\frac{w}{2})$ (for large $w$), corresponding to a hard wedge centered in $y$. Everywhere in this paper $\equiv$ means equivalent in law. The additive constant $\ln(\frac{w}{2})$ is necessary for regularization, but we will ignore below all time-independent constants.

More generally, for droplet initial conditions, the multi-point correlation of the field $h(x,t)$ is believed to converge ps-npoint ; reviewCorwin ; CorwinKPZensemble to the ones of the Airy2 process ${\cal A}_{2}(\hat{x})$ spohn2000 ; ferrariAiry with, in our units 222In several works, e.g. reviewCorwin ; ps-npoint ; QuastelSupremumAi2 , the dimensionless equation is chosen by setting $\nu=\frac{1}{2}$, $\lambda_{0}=1$ and $D=1$. This is equivalent to only a change of the time, i.e. it corresponds to the choice $t^{*}=2$ (i.e. $t^{\prime}=2t$ where $t$ denotes the time here and $t^{\prime}$ the time there).:

[TABLE]

where ${\cal A}_{2}(0)\equiv\chi_{2}$. Here $\simeq$ means in law, as a process as $x$ is varied. The process ${\cal A}_{2}(\hat{x})$ is stationary, i.e. statistically translationally invariant in $\hat{x}$, and well characterized: its correlations can be expressed in terms of, larger, Fredholm determinants in terms of the so-called extended Airy kernel. More generally, at large time

[TABLE]

where $h(x,t|y,0)$ is the droplet solution with arbitrary endpoints (8). In terms of processes, this equivalence is only valid at either fixed $y$ or fixed $x$. The process as $(x,y)$ are both varied is called the Airy sheet and is not yet characterized.

The second standard result is for the flat initial condition $h(x,t=0)=0$. There it was found we-flat ; we-flatlong that :

[TABLE]

where $F_{1}(s)$ is the GOE Tracy Widom (cumulative) distribution. It is expressed as a Fredholm determinant

[TABLE]

In that case, it is believed that the joint distribution of the heights $\{h_{flat}(x,t)\}_{x}$ is governed by the so-called Airy1 stationary process ${\cal A}_{1}(u)$:

[TABLE]

where ${\cal A}_{1}(0)=\frac{1}{2}\chi_{1}$. For definition and normalizations of the Airy1 process see e.g. Ref. Airy1TASEP ; ferrariAiry ; QuastelSupremumAi2 .

Note that there is a connection between these results. Indeed from the definition one expects, in the large time limit:

[TABLE]

where we have used that the sets $\{h(x,t|y,0)\}\equiv\{h(y,t|x,0)\}$ are statistically equivalent and that, since height fluctuations grow as $t^{1/3}$, the integral is dominated by its maximum. Since one can shift $\hat{y}$ by $\hat{x}$, the maximum of the Airy2 process minus a parabola is given by the Airy1 process at one point

[TABLE]

as proved in QuastelSupremumAi2 .

II.3.2 Main results of the present work

Let us summarize some of the results of the present work, more results, i.e. equivalent kernels, various limits, comparison with known results, and more cases, are presented in Section IV and Appendix D.

Here we obtain that, for the various initial conditions detailed in Section II.2, the following CDF is given in the limit of large time by a Fredholm determinant

[TABLE]

where the kernel $K$ takes the following forms in the various cases. We need to define the function

[TABLE]

where the second expression is only valid for $w>0$, while the first is valid for any real $w$. We find:

wedge initial condition ($a_{L}=a_{R}=0$)

[TABLE]

which interpolates between flat ($\hat{w}_{L,R}\to 0^{+}$) and droplet ($\hat{w}_{L,R}\to+\infty$) initial conditions, and contains the half-flat (crossover ${\cal A}_{2\to 1}$) as a special case for $\hat{w}_{L}=0^{+}$, $\hat{w}_{R}=+\infty$. 2. 2.

wedge-Brownian initial condition ($a_{L}=0$, $a_{R}=1$)

[TABLE]

which for $\hat{w}_{L}=+\infty$ reproduces the half-Brownian case. In the limit $\hat{w}_{L},\hat{w}_{R}\to 0^{+}$ we obtain the flat to stationary transition kernel given in (IV.2.2), which is one main result of this paper. 3. 3.

Brownian-Brownian initial condition ($a_{L}=1$, $a_{R}=1$)

[TABLE]

which, as is shown in Section IV reproduces the known result for the stationary case, although in a compact form (as a single Fredholm determinant) to our knowledge not presented before. 4. 4.

Finally we display the result for the step initial condition, here for simplicity for $\hat{w}_{L,R}=0$ and $a_{L,R}=0$, i.e. a flat initial condition plus a (descending) step of amplitude $\Delta>0$. For the large time limit to be non-trivial we must scale the step size as $t^{1/3}$, hence we define

[TABLE]

as the quantity kept fixed in the large time limit. In practice, since the KPZ equation has been derived, and is valid, only for small height gradients $\partial_{x}h\ll 1$, we can think of smoothing the step over a scale $\delta x\sim t^{a}$. The condition of small gradient only requires $a\geq 1/3$, and we need $a<2/3$ for our result to hold (equivalent to $t\gg\tau\gg 1$ in the formulation (2)). With this scaling, the kernel reads

[TABLE]

with however replacing the projector $P_{\sigma}\to P_{\sigma-\hat{\Delta}}$ in (23), see Eq. (152) and (IV.4) for more details. The generalization to arbitrary slopes $\hat{w}_{L,R}>0$ is given in the Appendix, equation (207).

Finally the result for a step on top of the Brownian-Brownian initial condition is given in (209). The result for a step on top of the wedge-Brownian initial condition is given in (211) and (212) (and includes the flat to stationary plus a step for $\hat{w}_{L,R}=0^{+}$).

II.4 Generating functions

To obtain these results we will define and calculate some generating functions. For notational convenience we introduce a second set of rescaled parameters

[TABLE]

where the parameter $\lambda$ was introduced in Ref. we ; dotsenko ; we-flat . As in these works we define the standard generating function

[TABLE]

where $Z(x,t)=e^{h(x,t)}$ and the second equality is only formal, as always in the RBA method for the continuum KPZ equation, since the sum is a divergent series. Examination of this series, however, will allow to obtain (or conjecture) the first average. In the large time limit it identifies with the CDF of the rescaled height

[TABLE]

In view of the initial condition (41) it is natural to split, in each realization of $(\eta,B_{L},B_{R})$, the DP partition sum into the set of paths starting at $x$ and ending either left or right of $y=0$, as

[TABLE]

with $Z_{0}(y)=Z(x,t=0)$, and to introduce the corresponding generalized generating function:

[TABLE]

the standard generating function being recovered for equal arguments

[TABLE]

This generalized generating function allows to study the initial condition (10) in presence of a step, which can be written as

[TABLE]

Hence the standard generating function for this problem, denoted $g_{\lambda}^{\Delta}(s)$, can be expressed as

[TABLE]

III Replica calculations

III.1 Averaging and quantum mechanics

The initial condition for the DP partition sum $Z(x,t)$ is:

[TABLE]

The solution of the SHE with this initial condition can be written:

[TABLE]

and we will calculate its positive integer moments with respect to the joint measure on $\eta$ and $(B_{L},B_{R})$, denoted here more explicitly by overline $\overline{\cdots}^{\eta}$ and bracket $<..>_{B_{L},B_{R}}$ respectively

[TABLE]

Since we have chosen $\eta$ and $B_{L},B_{R}$ to be independent it can be written as

[TABLE]

Let us recall the STS symmetry. Using appendix A of Ref. we-flatlong one easily sees that for all $n$

[TABLE]

equivalently

[TABLE]

The STS symmetry thus also fixes how the generating function depends on some combination of variables:

[TABLE]

As is now well known kardareplica ; bb-00 the $\eta$ average in the middle of (44) can be rewritten as the expectation value between initial and final states of the quantum-mechanical evolution operator associated to the attractive Lieb-Liniger (LL) Hamiltonian for $n$ identical particles ll :

[TABLE]

We can thus rewrite (44) in quantum mechanical notations:

[TABLE]

where $|x\dots x\rangle$ is the state will all particles at the same point $x$. Since this state is fully symmetric in exchanges of particles, only symmetric eigenfunctions will contribute and we can consider particles as bosons. The wavefunction of the initial replica state is:

[TABLE]

where here and below coordinate multiplets are denoted by capital letters, e.g. $Y\equiv y_{1},..y_{n}$. This state is also clearly symmetric in the replica, hence the argument about bosons can also be made with this state alone.

We now introduce the decomposition into eigenstates $\Psi_{\mu}$ and eigenenergies $E_{\mu}$ of $H_{n}$ and rewrite the moment as a sum over eigenstates

[TABLE]

where we have used that ${\cal Z}_{n}$ is real, and for convenience we will work with the second (i.e. complex conjugate) expression.

We can now use the explicit form of the eigenfunctions known from the Bethe ansatz ll . They are parameterized by a set of rapidities $\mu\equiv\{\lambda_{1},..\lambda_{n}\}$ which are solution of a set of coupled equations, the Bethe equations (see below). The eigenfunctions are totally symmetric in the $x_{\alpha}$, and in the sector $x_{1}\leq x_{2}\leq\dots\leq x_{n}$, take the (un-normalized) form

[TABLE]

They can be deduced in the other sectors from their full symmetry with respect to particle exchanges. The sum runs over all $n!$ permutations $P$ of the rapidities $\lambda_{\alpha}$. The corresponding eigenenergies are $E_{\mu}=\sum_{\alpha=1}^{n}\lambda_{\alpha}^{2}$. In the formula (51) we first need:

[TABLE]

and second, we need the overlap. Since both states are symmetric, their overlap can be rewritten as:

[TABLE]

using the explicit form of the Bethe eigenstates.

Introducing the numbers of replica $n_{L,R}$ on each side of $y=0$, it can be expressed as:

[TABLE]

with $G^{L}_{0,w,a}=G^{R}_{0,w,a}=1$. We have defined the integrals over the left and right half-axis

[TABLE]

We have taken advantage that $B_{L}$ and $B_{R}$ are independent and of the factorized form of each term in the wave-function in each sector. To evaluate these blocks we now use the following averages over a one-sided Brownian, valid for ordered coordinates, as indicated:

[TABLE]

and the integration identities, valid in the domains where the integrals converge:

[TABLE]

It leads to:

[TABLE]

Note that:

[TABLE]

For our two ”solvable” cases, $a=0,1$ a ”miracle” occurs upon performing the summation over the permutations, leading to a factorized form SasamotoHalfBrownReplica ; we-flat ; we-flatlong

[TABLE]

where we have introduced two new functions which depend only on the set of rapidities, not on their order. They obey now

[TABLE]

Note that these miracle identities allow to obtain simple expressions for the terms where either $n_{L}$ or $n_{R}$ is zero in (55) but (a priori) not for the general term since there are then permutations which exchange rapidities in the left and right groups of rapidities. These simpler cases were used to obtain solutions for the half-flat and half-Brownian initial conditions SasamotoHalfBrownReplica ; we-flat ; we-flatlong (formally obtained by taking one of the slopes to infinity).

In the present case one does not seem able to proceed further without specifying the eigenfunctions 333except for the case $a_{L}=a_{R}=1$, i.e. Brownian on both sides where there is a further miracle identity SasamotoStationary .. We now recall that in the large $L$ limit one can work with string eigenstates. These possess specific properties which allow to obtain explicit expressions. This is based on combinatorial properties which were first claimed by Dotsenko dotsenkoGOE , and used by him in the case of the wedge (mostly for infinitesimal $w_{L,R}$). We will re-formulate, check, and slightly generalize these combinatorial identities and apply them to other cases.

III.2 Strings and combinatorial identities

Let us recall the spectrum of $H_{n}$ in the limit of infinite system size, i.e. the rapidities solution to the Bethe equations m-65 . A general eigenstate is built by partitioning the $n$ particles into a set of $1\leq n_{s}\leq n$ bound states called strings each formed by $m_{j}\geq 1$ particles with $n=\sum_{j=1}^{n_{s}}m_{j}$. The rapidities associated to these states are written as

[TABLE]

where $k_{j}$ is a real momentum, the total momentum of the string being $K_{j}=m_{j}k_{j}$. Here, $a=1,...,m_{j}$ labels the rapidities within the string $j=1,\dots n_{s}$. We will denote $|\mu\rangle\equiv|{\bf k},{\bf m}\rangle$ these strings states, labelled by the set of $k_{j},m_{j}$, $j=1,..n_{s}$. Here and below the boldface represents vectors with $n_{s}$ components.

Inserting these rapidities in (52) leads to the Bethe eigenfunctions of the infinite system, and their corresponding eigenenergies:

[TABLE]

We have separated a trivial part of the energy, which can be eliminated by defining

[TABLE]

i.e. leading to a simple shift in the KPZ field. We will implicitly study in the remainder of the paper $\tilde{\cal Z}_{n}$, $\tilde{Z}(x,t)$ and $\tilde{h}(x,t)$ but will remove the tilde in these quantities for notational simplicity (as already mentioned in the introduction).

The formula for the norm of the string states has been obtained as cc-07 :

[TABLE]

so that the formula (51) for the moments becomes for $L\to+\infty$ (provided all limits exist)

[TABLE]

where the second sum is over the set of partitions, denoted $(m_{1},..m_{n_{s}})_{n}$, of the integer $n=\sum_{j=1}^{n_{s}}m_{j}$ into $n_{s}$ parts, with each $m_{j}\geq 1$.

It remains to calculate the overlap, formula (55). If the states are strings, the sum over permutations can be performed, using a general combinatoric identity which is detailed in the Appendix. The result is:

[TABLE]

for any string state. The sum corresponds to all possible ways to split rapidities in two groups associated to particles on $x<0$ (left) and $x>0$ (right). The combinatoric factor ${\cal G}$ is a complicated product of Gamma functions and given in the Appendix (A). It coincides with the one obtained by Dotsenko dotsenkoGOE . It will not play a role in the following as it will be set to unity in the large time limit. The $\tilde{H}^{L,R}$ terms are obtained by evaluating $H^{L}$ and $H^{R}$ on the two following complementary sets of rapidities

[TABLE]

where the first set contains the $n_{L}$ rapidities on the left, and the second the $n_{R}$ on the right, with the following notation for the string rapidities in the two groups

[TABLE]

The functions $\tilde{H}^{L,R}$ are thus only functions of the set of $\{k_{j},m_{j}^{L},m_{j}^{R}\}_{j=1,..n_{s}}$, equivalently $\{k_{j},m_{j}^{L},m_{j}\}_{j=1,..n_{s}}$ or $\{k_{j},m_{j},m_{j}^{R}\}_{j=1,..n_{s}}$, since $m_{j}^{L}+m_{j}^{R}=m_{j}$. They satisfy the symmetry relations

[TABLE]

III.3 calculation of the factors $\tilde{H}$

We can now evaluate the functions $\tilde{H}^{L,R}$ by injecting the string rapidities (72), into the formula (61) and (62), according to the rule (71). This leads to expressions involving Pochhammer symbols, equivalently Gamma functions. We have:

[TABLE]

The single string factors are:

[TABLE]

On the first line it reproduces, for $m_{R}=0$, the one obtained in Ref. we-flat (Section 5.1) for the half-flat initial condition. The factors involving two strings are:

[TABLE]

and $D^{w_{L},1}_{m_{j}^{L},m_{j}^{R},k_{i},m_{j^{\prime}}^{L},m_{j^{\prime}}^{R},k_{j^{\prime}}}=1$, with $z_{jj^{\prime}}=2w_{L}+ik_{j}+ik_{j^{\prime}}$. In the Brownian case there is thus no inter-string factors. For the half-flat case, (77) reproduces the one of Ref. we-flat (Section 5.1) when $m_{j}^{R}=m_{j^{\prime}}^{R}=0$.

III.4 General formula for the moments and generating function

Putting all together we thus obtain the general formula for the moments

[TABLE]

Note that in this sum the term with fixed $n_{L},n_{R}$ has a simple interpretation. Consider (36), (37), where in each realization of $(\eta,B_{L},B_{R})$, the DP partition sum is splitted into the set of paths starting at $x$ and ending either left or right of $y=0$, The moments then split as:

[TABLE]

Hence, by simple identification of the terms with fixed $n_{L},n_{R}$ in (78) and multiplication by the factor $\frac{n_{L}!n_{R}!}{n!}$ we obtain an expression for each joint moment $\overline{(Z^{L})^{n_{L}}(Z^{R})^{n_{R}}}$.

Let us come back to the generating function (37). The sums over the variables $m,m_{L},m_{R}$ now become free summations leading to

[TABLE]

with

[TABLE]

Although this is an exact and explicit expression, apart from the case $a_{L}=a_{R}=1$, it is unclear how to handle it for arbitrary time. We thus now turn to the large time limit.

III.5 large time limit

In the large time limit we will assume that one can set the product of factors $D$ and ${\cal G}$ to unity. This is of course a highly non-trivial and radical assumption, however it is justified a posteriori by the results. It will be checked in all cases where the solution is known by other means. This procedure follows what has been done in other works, where it was also checked against other methodsdotsenkoGOE ; PLDCrossoverDropFlat ; ps-2point ; dotsenko2pt ; Spohn2ptnew .

III.5.1 determinantal form

Let us first obtain a closed expression once these factors are set to unity, and take the large $\lambda$ limit in a second stage:

[TABLE]

We now use the standard determinant double-Cauchy identity:

[TABLE]

and perform the rescaling $k_{j}\to k_{j}/\lambda$. Denoting $\tilde{x}=x/\lambda^{2}$ we obtain:

[TABLE]

We now perform the Airy trick, i.e. use the representation $e^{\frac{1}{3}\lambda^{3}m^{3}}=\int dyAi(y)e^{\lambda my}$ to obtain

[TABLE]

Using standard manipulations we ; dotsenko the partition sum at fixed number of string $n_{s}$ can thus be expressed itself as a determinant:

[TABLE]

with the Kernel:

[TABLE]

where the $S$ factors are given explicitly in (76). The generating function is thus a Fredholm determinant:

[TABLE]

where, again, this expression is valid as soon as the factors $D$ and ${\cal G}$ are set (arbitrarily) to unity.

We must now study the function $\phi_{\lambda}(k,y_{L},y_{R})$ in the large time limit $\lambda\to+\infty$.

III.5.2 large time limit

We first rewrite:

[TABLE]

and we use the Mellin-Barnes identity:

[TABLE]

where $C=\kappa+i\mathbb{R}$, $-1<\kappa<0$, valid provided $f(z)$ is meromorphic, with no pole for $z>\Re(\kappa)$, and sufficient decay at infinity. It allows to rewrite (for $2w_{L,R}+\kappa>0$)

[TABLE]

Here the analytic continuation $f(m)\to f(z)$ has been performed using the second expression in (76) as

[TABLE]

We now rescale $z_{L,R}\to z_{L,R}/\lambda$, and we study the large time limit $\lambda\to+\infty$. We first recall the definition of the rescaled drifts:

[TABLE]

and we use that for $a=0,1$:

[TABLE]

as can be seen from (98). Thus we obtain the infinite $\lambda$ limit in the form of a double contour integral:

[TABLE]

where $C^{\prime}=0^{-}+i\mathbb{R}$. The calculation of this integral is performed in Appendix B and the result is displayed in (184).

For now, we focus on the simpler generating function, i.e. we set $s_{L}=s_{R}=s$. In the infinite time limit it takes the form

[TABLE]

where $P_{0}(v)=\theta(v)$ is the projector on $[0,+\infty[$ and with the Kernel

[TABLE]

The function $\phi_{+\infty}(k,y):=\phi_{+\infty}(k,y_{L}=y,y_{R}=y)$ being obtained from the more general result (184) in the Appendix.

We now use Airy function identities in order to rewrite the result in terms of an alternative kernel. The calculation is performed in Appendix C. The final result is:

[TABLE]

which, we note depends only on $\sigma$ and the combinations $\hat{w}_{L}+\hat{x}$ and $\hat{w}_{R}-\hat{x}$ (and their sum), as required by the STS symmetry (47).

IV Results for the various crossover kernels

We now discuss in details the results for the various initial conditions. We give both the result in the first form (102), with kernel $M_{s}$ from (103)

[TABLE]

naturally expressed in the variables $\tilde{x},\tilde{w}_{L,R}=2^{-2/3}t^{1/3}w_{L,R}$, and the second, equivalent form of the result (105), with kernel $K_{\sigma}$ and $K$ from (107)

[TABLE]

naturally expressed in the variables $\sigma,\hat{x},\hat{w}_{L,R}=t^{1/3}w_{L,R}$.

IV.1 the wedge initial condition

Let us start with the wedge initial condition (41) with $a_{L}=a_{R}=0$.

IV.1.1 first form of the wedge kernel

From (103) we find

[TABLE]

Let us discuss several limits.

Half-flat initial condition and GUE-GOE crossover: For $\tilde{w}_{R}\to+\infty$ one recovers the Kernel for the half-flat case obtained in Ref. PLDCrossoverDropFlat (formula (80-81) there). As shown there it interpolates between the GOE (flat) for $\tilde{x}\to-\infty$ and the GUE (droplet) kernels for $\tilde{x}\to+\infty$.

Symmetric wedge: In that case one chooses $\tilde{w}_{L}=\tilde{w}_{R}=w$. We obtain

[TABLE]

This kernel also provides an interpolation from GUE (droplet) to GOE (flat) as $\tilde{w}$ is decreased from $+\infty$ to [math]. The two limits are particularly immediate on that form of the kernel. For $\tilde{w}\to+\infty$ one has:

[TABLE]

This is identical to the GUE kernel in the form given in we . In the other limit $\tilde{w}\to 0^{+}$ we can replace

[TABLE]

leading to:

[TABLE]

which is the simplest form of the GOE kernel.

IV.1.2 second form of the wedge kernel

The second form of the wedge kernel reads:

[TABLE]

In the limit $\tilde{w}_{R}\to+\infty$ one recovers the Kernel for the half-flat case in the second form obtained in PLDCrossoverDropFlat (formula (89-91) there), namely

[TABLE]

which, as discussed there, is equivalent to the result of Ref. BorodinAiry2to1 for TASEP. As shown there it interpolates between the GOE (flat) for $\hat{x}\to-\infty$ and the GUE (droplet) kernels for $\hat{x}\to+\infty$, i.e. it is the (one-point) kernel associated to the ${\cal A}_{2\to 1}$ interpolation process.

More generally in the double limit $(\hat{w}_{L},\hat{w}_{R})\to(0^{+},0^{+})$ we obtain, using another Airy identity:

[TABLE]

Hence using that $\sigma-\hat{x}^{2}=2^{-2/3}s$:

[TABLE]

Under a similarity transformation this is equivalent to the GOE kernel:

[TABLE]

IV.2 the wedge-Brownian initial condition

Let us now consider now the wedge-Brownian initial condition (41), with $a_{L}=0$ and $a_{R}=1$. This case contains the flat to stationary crossover as a limit, see below.

IV.2.1 first form of the kernel

From (103) we find:

[TABLE]

IV.2.2 second form of the kernel

The second form of the kernel reads:

[TABLE]

where we have performed the change of variable $y\to y/2$ in the second term. Note that the second integral is convergent only for $\hat{w}_{R}-\hat{x}>0$. It is however easily extended to arbitrary values (see below).

Half-Brownian limit: in the limit $\hat{w}_{L}\to+\infty$ one should recover the half-Brownian initial condition. In that limit

[TABLE]

Note that the second integral is convergent only for $\hat{w}_{R}-\hat{x}>0$. To obtain a more general expression, we can use the identity, valid for $u>0$:

[TABLE]

and replace

[TABLE]

and expression where now the integrals are convergent for any $\hat{w}_{R}-\hat{x}$ and which coincides with the asymptotic large time formula (2.23) in Ref. SasamotoHalfBrownReplica (the correspondence is that $X,\gamma_{t}$ there are $X=\hat{x}-\hat{w}_{R}$, $\gamma_{t}=t^{1/3}$). Thus the above replacement (127) is legitimate (it can in fact be shown also from the first form of the kernel, repeating the calculation of Appendix C) and we will use it repeatedly in the following.

We can now go back to the general case of the wedge-Brownian initial condition (129) and note that it can be written as the sum of the half-flat kernel (which interpolates between GUE and GOE) and a projector

[TABLE]

where

[TABLE]

and $K^{\rm half-flat}(v_{i},v_{j})$ is given in (119).

Flat to stationary crossover: It is now possible to consider the limit $\hat{w}_{L},\hat{w}_{R}\to 0^{+}$. One obtains

[TABLE]

which is the main result of this paper. It has the form of the ${\cal A}_{2\to 1}$ transition kernel plus a projector.

IV.3 the Brownian-Brownian initial condition

Consider now the Brownian-Brownian initial condition (41), with $a_{L}=a_{R}=1$, i.e. a double sided Brownian initial condition.

IV.3.1 first form of the kernel

From (103) we find:

[TABLE]

IV.3.2 second form of the kernel

The second form of the kernel reads:

[TABLE]

where we have defined

[TABLE]

where the second form is valid for $w>0$, while the first one is valid for arbitrary $w$ (see discussion above).

We now show that this result is equivalent to the result of Ref. SasamotoStationary in the large time limit. The notations of that paper are $X=\hat{x}$, $\gamma_{t}=t^{1/3}$, $v_{\pm}=w_{R,L}$ hence $\omega_{\pm}=v_{\pm}\gamma_{t}=\hat{w}_{R,L}$ (with $\alpha=1$ in our units $\nu=1,D=2,\lambda_{0}=2$). The CDF of the height was obtained in formula (6.21-22) at at large time (correcting the misprint $X\to-X$ there):

[TABLE]

To show that they are the same, let us first express explicitly the derivative

[TABLE]

we have used that ${\rm Det}(1-P_{\sigma}BP_{\sigma})={\rm Det}(1-P_{0}B_{\sigma}P_{0})$ where $B_{\sigma}(v_{1},v_{2})=B(v_{1}+\sigma,v_{2}+\sigma)$. To obtain $\partial_{\sigma}B_{\sigma}$ we calculate the following derivatives

[TABLE]

where we used integration by parts. Hence we obtain

[TABLE]

Now we note that it can be written as a product

[TABLE]

where we have defined

[TABLE]

Hence $\frac{1}{\hat{w}_{R}+\hat{w}_{L}}\partial_{\sigma}B_{\sigma}(v_{1},v_{2})$ is a projector, which implies that (138) can be rewritten as

[TABLE]

since one can check that our Kernel (134) can be written as:

[TABLE]

As is discussed in Ref. SasamotoStationary the expression (136) is equivalent to the one in Theorem 5.1. of Ref. ImamuraPNG derived for the PNG model with external sources. The relation between (136) and the result of Baik and Rains in Ref. png (in terms of the solution of a Painleve II equation) is discussed in Proposition 5.2 of Ref. SasamotoStationary . The result also agrees with the one for stationary TASEP spohnTASEP and a rigorous derivation was given in BCFV . Note that in Ref. SasamotoStationary the solution is given for arbitrary time, which is possible in that case. This provides a test of our more direct (but more empirical) method to obtain directly the large time limit.

IV.4 Adding a step to the initial condition

IV.4.1 first form of the kernel

Consider now the step initial condition (10). As discussed in previous sections, to obtain the solution for that case, in the large time limit, we need to calculate the generalized generating function in the large time limit

[TABLE]

where $\tilde{\Delta}=\Delta/\lambda$. We will specify to $a_{L}=a_{R}=0$, i.e. step initial condition on top of the wedge. The solutions for the two other cases, the step plus half-Brownian (or step on top of flat to stationary), and step on top of two-sided Brownian are given in Appendices D.5 and D.4 respectively.

From (92), (94) and the result (184) in Appendix B we can write that

[TABLE]

Let us give the result for $\tilde{\Delta}>0$, i.e. $s_{R}>s_{L}$

[TABLE]

with

[TABLE]

at this stage we have also kept arbitraty slopes $\tilde{w}_{L,R}$.

IV.4.2 second form of the kernel

We now obtain the second form for the result. The details are given in Appendix D and D.3. We find

[TABLE]

with the kernel

[TABLE]

The generalization to arbitrary slopes $\hat{w}_{L,R}>0$ is given in the Appendix, equation (207).

Note that (152) can also be written, denoting $\sigma^{\prime}=\sigma-\Delta$, as

[TABLE]

Hence $\sigma^{\prime}$ measure the fluctuations w.r.t the height level of the step on the left ($x<0$). Thus, if $\hat{\Delta}\to+\infty$ the step size becomes infinite and the height level on the right goes to $-\infty$ (relatively to the left). Thus one must find the half-flat kernel, and indeed one can check that the second and fourth term in (153) vanish in that limit, i.e. $K_{\hat{\Delta}\to+\infty}(v_{i},v_{j})\to K^{\rm half-flat}(v_{i},v_{j})$, as given in (119)

In the limit $\hat{\Delta}\to 0$ the first two terms in (153) cancel and one finds

[TABLE]

Hence

[TABLE]

with $s=2^{2/3}\sigma$ and we recall $K^{GOE}(v_{i},v_{j})=Ai(v_{i}+v_{j})$. Hence we recover the result for the flat initial condition (18).

V Conclusion

In conclusion we have used the replica Bethe ansatz method to study the distribution of the the scaled interface height at one space time point in the 1D KPZ equation, with a set of initial conditions which are different on the negative and the positive half line. This set contains all standard crossover classes between respectively flat, droplet and stationary on each side, as well as in presence of slopes (i.e. drifts). The method also allows to add a step at the origin for each of these initial conditions. The slopes and step parameters, as well as the coordinate of the observation point, are properly scaled with time so that the result is non-trivial in the large time limit and interpolates between various classes of initial conditions, as they are varied. This generalizes our previous work on the crossover between flat and droplet. In all cases the one point CDF of the height can be expressed as a Fredholm determinant with various kernels depending on the parameters. All these expressions, although obtained starting from the KPZ equation, are conjectured to be universal for all models in the 1D KPZ class.

The method contains some heuristics, following previous works, as it assumes that in the large time limit, a decoupling occurs, so that some terms can be set to unity in the complicated sum over string eigenstates, allowing for an exact calculation. The calculation is performed by using, and further testing and extending, a combinatorics method introduced by Dotsenko. We test the validity of the method in cases where the answer is known, such as flat, droplet and their crossover, as well as Brownian and half Brownian. In these cases, it reproduces the known result, although sometimes naturally leading to new, equivalent, forms for the kernels. In all other cases, it produces some conjectures for the kernels. Among them, the flat to stationary crossover kernel is directly obtained. It would be interesting to confirm all the present results by different methods.

*Note added: * while this work was in the last stages of completion, we learned of the recent work of Quastel and Remenik QuastelRemenikParabola , who obtained a general formula for a very large class of initial conditions. Although these do not yet allow to average over random initial conditions (such as Brownian) it would be interesting, in the deterministic case, to compare their formula (obtained for Airy processes) and the present results (obtained starting from the KPZ equation). In an even more recent work KPZFixedPoint2 they prove the convergence to such formula starting from TASEP.

Acknowledgements.

I am grateful to A. Borodin, P. Calabrese, I. Corwin, P. Ferrari, J. Quastel and D. Remenik for useful discussions and pointing out several important references.

Appendix A General identity

A.1 Preliminaries

Consider the Bethe wave function (52), $\Psi_{\mu}(X)$, which is a symmetric function of its arguments, and which, for $x_{1}\leq..\leq x_{n}$ reads

[TABLE]

where $\lambda_{1},\cdots\lambda_{n}$ are the rapidities. Split the coordinates $x_{\alpha}$ into two groups, $N_{L}$ with $\alpha=1,\cdots,n_{L}$, i.e. $x_{1}\leq x_{2}\leq\cdots\leq x_{n_{L}}$ and $N_{R}$ with $\alpha=n_{L}+1,\cdots,n_{L}+n_{R}=n$, i.e. $x_{n_{L}+1}\leq\cdots\leq x_{n}$, a splitting which respects the constraint $x_{1}\leq\cdots\leq x_{n}$, i.e. such that $x_{a}\leq x_{b}$ for all $a\in N_{L}$ and $b\in N_{R}$. In (158), for each permutation $P$ a rapidity $\lambda_{P_{\alpha}}$ is associated to each coordinate $x_{\alpha}$, hence for each permutation $P$ a first $n_{L}$-uplet of rapidities $(\lambda_{P_{1}},\cdots,\lambda_{P_{n_{L}}})$ is associated to the group $N_{L}$ and a second $n_{R}$-uplet, $(\lambda_{P_{n_{L}+1}},\cdots,\lambda_{P_{n}})$ is associated to the group $N_{R}$.

In a number of applications one needs to calculate

[TABLE]

where here we will consider $F_{n_{L}}^{L}$ and $F_{n_{R}}^{R}$ to be arbitrary functions of $n_{L}$, respectively $n_{R}$, variables, with a priori no symmetry (i.e. functions of the $n_{L}$-uplet and $n_{R}$-uplet, respectively). This is the case for instance for the calculation of the overlap of $\Psi_{\mu}(X)$ with any other wave function which splits into a product over $N_{L}$ and $N_{R}$, see (55) as an example. Note that there we eventually sum over $n_{L},n_{R}=n-n_{L}$ but here we will consider the more general question of evaluation of (159) for any fixed $n_{L},n_{R}=n-n_{L}$.

Let us consider this question when the rapidities are strings. So consider now a Bethe state with $n_{s}$ strings specified by $k_{j},m_{j}$, $j=1,\cdots,n_{s}$, i.e. rapidities labeled as:

[TABLE]

In notations of the text such a state is denoted as $|{\bf k},{\bf m}\rangle$. As is well known, and clear from the definition (158), the only permutations $P$ which have a non vanishing amplitude $A_{P}$, are those such that for each string the intra-string order of increasing imaginary part is maintained. Hence if one is given the set of $2n_{s}$ integers $(m^{L}_{j},m^{R}_{j})$, $j=1,\cdots,n_{s}$:

[TABLE]

which specifies how many particles in each string belongs to each of the two groups, then one knows (bijectively) the two sets of rapidities which belong of each group. For instance one knows that the first set of rapidities is:

[TABLE]

and the second set is the complementary $\Lambda_{R}=\{\lambda_{j,r_{j}},j=1,..n_{s},r_{j}=m_{j}^{L}+1,..m_{j}\}$. To treat these two sets on equal footing, it is convenient to introduce the notation:

[TABLE]

Note that the sets are now specified, but that within each set, one still needs to sum over all possible orders, i.e. permutations. That is, to each of these two sets, one can associate $n_{L}!$ possible $n_{L}$-uplets (respectively $n_{R}!$ possible $n_{R}$-uplets) of rapidities.

Consider now the quantity defined by (159) for a given string state $|{\bf k},{\bf m}\rangle$. One can guess that the sum over $P\in S_{n}$ in (159) can now be made in two stages. In a first stage one fixes the ${\bf m}^{R}=\{m^{R}_{j}\}_{j=1,\cdots n_{s}}$, equivalently the ${\bf m}^{L}=\{m^{L}_{j}\}_{j=1,\cdots n_{s}}$, and perform the sum over permutations inside each set, and then, in a second stage sum over the variables ${\bf m}^{R,L}$. One takes advantage that one can factor $A_{P}$ as

[TABLE]

and one defines

[TABLE]

Clearly $H_{n_{L}}^{L}$ and $H_{n_{R}}^{R}$ are now fully symmetric functions of their arguments. One can then evaluate $H_{n_{L}}^{L}$ on the set $\Lambda_{L}$ and $H_{n_{R}}^{R}$ on the set $\Lambda_{R}$. One thus defines:

[TABLE]

Note that the functions on the left do not explicitly depend any more on the choice $n_{R},n_{L}$, they depend on this choice only via $n_{L}=\sum_{j}m_{j}^{L}$ and $n_{R}=\sum_{j}m_{j}^{R}$.

Let us now consider the last factor in (165) and evaluate it on the string state $|{\bf k},{\bf m}\rangle$

[TABLE]

where we have defined the function ${\cal G}$ and the factors $G$ can be expressed using Pochammer symbols $(x)_{n}=\Gamma(x+n)/\Gamma(x)$ as follows

[TABLE]

where we have replaced everywhere $m_{j}=m_{j}^{L}+m_{j}^{R}$, and $G_{j^{\prime}j}$ is obtained by simply exchanging all indices $j$ and $j^{\prime}$. The function ${\cal G}$ can thus be written in terms of Gamma functions

[TABLE]

Note that using the identity $\Gamma(1-x)=\pi/(\Gamma(x)\sin(\pi x))$ we can rewrite this function differently. One can check that for integers $m_{j}^{L,R}$ the factors containing the sinus functions all together simplify to unity. Hence the function $G$ can equivalently be written as

[TABLE]

which shows that the question of its analytic continuation to $m_{j}^{L,R}$ complex is non-trivial (non-unique). Indeed if one were to attempt calculations including this factor using Mellin Barnes formula, one could argue from the form (175) that the standard scaling at large time $k_{j}\to k_{j}/\lambda$, $m_{j}^{L,R}\to z_{j}^{L,R}/\lambda$ leads to ${\cal G}\to 1$, however on the second form such a property does not seem to hold.

A.2 Main identity

Our main result is the following general identity, for the evaluation of (159) for a given string state $|{\bf k},{\bf m}\rangle$, valid for any fixed $(n_{L},n_{R},n_{s},m_{j},k_{j})$ and arbitary functions $F_{n_{L}}^{L}$, $F_{n_{R}}^{R}$

[TABLE]

where the functions $\tilde{H}^{L}$, $\tilde{H}^{R}$ and ${\cal G}$ are given above. We have not attempted to prove this identity, but we have checked it using mathematica for a large set of values of the parameters $n_{R},n_{L},n_{s}$. Setting the functions $F_{n_{L}}^{L}$, $F_{n_{R}}^{R}$ to unity we have also checked the (quite non-trivial) ”normalisation identity”:

[TABLE]

which can be seen as an indentity involving Gamma functions. We have also checked that the above expression for ${\cal G}$ (once multiplied by its symmetric) is consistent with the formula given by Dotsenko dotsenkoGOE .

Appendix B Calculation of the auxiliary function $\phi_{+\infty}(k,y_{L},y_{R})$

To perform the integrals in (101) we expand the product, leading to four terms. We use the elementary integrals

[TABLE]

This allows to show, assuming everywhere ${\rm Re}(A)>0$:

[TABLE]

Taking a derivative w.r.t. $y_{R}$ one obtains:

[TABLE]

which allows to evaluate the two cross-terms in (101). We also need:

[TABLE]

and taking two derivatives we obtain:

[TABLE]

Putting all together, denoting $A_{L}=2\tilde{w}_{L}+2ik$ and $A_{R}=2\tilde{w}_{R}-2ik$ and slightly simplifying using that $(a_{L},a_{R})\in\{0,1\}^{2}$ we obtain from (101)

[TABLE]

If we set $y_{L}=y_{R}=y$ we obtain the formula (104) in the text.

Appendix C Airy function identities and second form of the Kernel

We use the Airy function identities (see e.g. Section 9 in Ref. PLDCrossoverDropFlat and references therein)

[TABLE]

where we assumed $\tilde{w}>0$.

Consider now the expression (103) for the kernel $M_{s}(v_{i},v_{j})$ and enumerate the terms upon expanding the products in (104). In the same order as they appear there, we use the above identities as follows. In the first four terms we use the first identity. In the first term we use $a=v_{i}+(y+s)/2$ and $b=v_{j}+(y+s)/2$, in the second term we use $a=v_{i}+s/2+y(1-a_{L}+a_{R})/2$ and $b=v_{j}+s/2+y(1+a_{L}-a_{R})/2$, in the third term we use $a=v_{i}+s/2$ and $b=v_{j}+s/2+y$, in the fourth term we use $a=v_{i}+s/2+y$ and $b=v_{j}+s/2$. In the last three terms we use the same $a,b$ as in the first term, and use respectively the second, third and first identities. This gives $M_{s}(v_{i},v_{j})=e^{\frac{\tilde{x}}{4}(v_{j}-v_{i})}\tilde{M}_{s}(v_{i},v_{j})$ with:

[TABLE]

In the final Fredholm determinant the common factor $e^{\frac{\tilde{x}}{4}(v_{j}-v_{i})}$ can be discarded, since ${\rm Det}[I+P_{0}M_{s}P_{0}]={\rm Det}[I+P_{0}\tilde{M}_{s}P_{0}]$.

We now rescale $y\to 2^{2/3}y$ in the first and second term, $y\to 2^{-1/3}y$ in the third and fourth term, and $r\to 2^{5/3}y$ in the last two terms, and we use the similarity transformation $M_{s}(v_{1},v_{2})=-2^{1/3}K_{\sigma}(2^{1/3}v_{1},2^{1/3}v_{2})$ and we obtain the result (105,107) displayed in the text.

Appendix D Generalized kernels and step initial conditions

D.1 General case: first form of kernel

Here we obtain the kernels associated to the generalized generating function (37). We start with the first form (92)-(94)

[TABLE]

To express $\phi_{+\infty}(k,y-s_{L},y-s_{R})$ let us insert $y_{L}=y-s_{L}$ and $y_{R}=y-s_{R}$ in (184). We obtain

[TABLE]

with $A_{L}=2\tilde{w}_{L}+2ik$ and $A_{R}=2\tilde{w}_{R}-2ik$.

D.2 General case: second form of kernel

We now rewrite the kernel using the Airy function identities given in the previous section. This gives $M_{s_{L},s_{R}}(v_{i},v_{j})=e^{\frac{\tilde{x}}{4}(v_{j}-v_{i})}\tilde{M}_{s}(v_{i},v_{j})$ with:

[TABLE]

a sum of ten terms. The identities are used with $a=v_{i}+\frac{y-R}{2}$ and $b=v_{j}+\frac{y+R}{2}$ where $R$ is as follows. In term $1$ of (200) (which comes from term $1,2$ of (199)) we use $R=0$ and identity $1$, then we list similarly: term $2$ (term 7) $R=y(a_{L}-a_{R})+s_{R}(1-a_{L})-s_{L}(1-a_{R})$ identity $1$; term $3$ (term 3) $R=y-s_{L}$ identity $1$; term $4$ (term 5) $R=s_{R}-y$ identity $1$; term $5$ (first piece of term 4) $R=0$ identity $2$; term $6$ (second piece of term 4) $R=s_{R}-s_{L}$ identity $2$; term $7$ (first piece of term 6) $R=0$ identity $3$; term $8$ (second piece of term 6) $R=s_{R}-s_{L}$ identity $3$; term $9$ (first piece of term 8) $R=\frac{1}{2}(s_{R}-s_{L})$ identity $1$; term $10$ (second piece of term 8) $R=\frac{1}{2}(s_{R}-s_{L})$ identity $1$;

We now want rewrite the generating function using the second kernel

[TABLE]

and we define

[TABLE]

where we use that $2^{1/3}\frac{\tilde{x}^{2}}{32}=\hat{x}^{2}$. We obtain

[TABLE]

To obtain this it it more convenient to first define $\sigma_{L,R}=2^{-2/3}s_{L}$. Then, in all terms we have performed a similarity transformation $2^{1/3}v_{i},2^{1/3}v_{j}\to v_{i},v_{j}$ which multiplies the kernel by $2^{-1/3}$. In terms $1-4$ we have rescaled $y\to 2^{2/3}y$, in terms $5-8$ we have integrated over the delta functions, then changed variable $r=2^{5/3}y$. The last step was to make the substitution in the resulting formula, $\sigma_{L,R}\to\sigma_{L,R}-\hat{x}^{2}$ and, simultaneously change $y\to y-\hat{x}^{2}$ but only in terms $1-4$.

D.3 Step on top of the wedge initial condition

We now want to apply this formula to the step initial conditions. From the text we have

[TABLE]

where $\tilde{\Delta}=\Delta/\lambda$. This can be rewritten using the above results as

[TABLE]

where we recall $\hat{\Delta}=\Delta/t^{1/3}$.

Let us specify to the case $a_{L}=a_{R}=0$, which represents the wedge plus a step. With no loss of generality, let us restrict to the case $\sigma_{R}>\sigma_{L}$, i.e. $\Delta>0$. The kernel then can be written

[TABLE]

We can now consider the limit $\hat{w}_{L,R}\to 0^{+}$, which leads to the well defined (trace class) kernel given in (153).

D.4 Step on top of the Brownian-Brownian initial condition

Let us specify to the case $a_{L}=a_{R}=1$, which represents the two-sided Brownian (plus drifts) initial condition plus a step. With no loss of generality, let us restrict to the case $\sigma_{R}>\sigma_{L}$, i.e. $\Delta>0$. The kernel then can be written

[TABLE]

It can be rewritten is a more generally valid form

[TABLE]

using the functions ${\cal B}_{w}(v)$ defined in (135). On this form it is apparent that as $\hat{\Delta}\to 0$ the kernel converges to the one for the Brownian-Brownian case (134). In the opposition limit $\hat{\Delta}\to+\infty$ we see that it converges as it should to the half-Brownian limit (128) (upon exchange of left and right, and kernel transposition).

D.5 Step on top of the wedge-Brownian initial condition

Let us specify to the case $a_{L}=0$, $a_{R}=1$, which represents an initial condition which is flat on the left (with a drift), Brownian on the right (with a drift) and, on top of it, a step. We obtain, from (204)

[TABLE]

Now we must distinguish the two cases $\Delta>0$ and $\Delta<0$.

Let us start with $\Delta>0$ (downward step), i.e. the case $\sigma_{R}>\sigma_{L}$. The kernel then can be written

[TABLE]

For $\hat{\Delta}=+\infty$ it goes as expected to the half flat kernel given in (119). For $\hat{\Delta}=0^{+}$ is goes to the wedge-Brownian kernel given in (129).

Let us consider now $\Delta<0$ (upward step), i.e. the case $\sigma_{R}<\sigma_{L}$. The kernel then can be written

[TABLE]

In the limit $\hat{\Delta}\to-\infty$ we see that it converges as it should to the half-Brownian limit (128), and for $\hat{\Delta}=0^{-}$ is goes to the wedge-Brownian kernel given in (129), being of course continuous at $\hat{\Delta}=0$.