Exponential convergence to equilibrium in supercritical kinetically constrained models at high temperature
Laure Mar\^ech\'e
TL;DR
This paper proves that supercritical kinetically constrained models in one and two dimensions rapidly reach equilibrium at high temperatures, extending understanding of their convergence behavior.
Contribution
It provides the first non-perturbative proof of exponential convergence to equilibrium for general supercritical KCMs in low dimensions at high temperature.
Findings
Convergence to equilibrium is exponential when $q$ is close to 1.
Results apply to a broad class of supercritical KCMs in 1D and 2D.
Initial configurations with Bernoulli distribution converge rapidly.
Abstract
Kinetically constrained models (KCMs) were introduced by physicists to model the liquid-glass transition. They are interacting particle systems on in which each element of can be in state 0 or 1 and tries to update its state to 0 at rate and to 1 at rate , provided that a constraint is satisfied. In this article, we prove the first non-perturbative result of convergence to equilibrium for KCMs with general constraints: for any KCM in the class termed "supercritical" in dimension 1 and 2, when the initial configuration has product law with , the dynamics converges to equilibrium with exponential speed when is close enough to 1, which corresponds to the high temperature regime.
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Taxonomy
TopicsTheoretical and Computational Physics · Stochastic processes and statistical mechanics · Material Dynamics and Properties
Exponential
convergence to equilibrium in supercritical kinetically constrained models at high temperature
Laure Marêché
Laure Marêché, LPSM UMR 8001, Université Paris Diderot, Sorbonne Paris Cité, CNRS, 75013 Paris, France
I acknowledge the support of the ERC Starting Grant 680275 MALIG
Abstract: Kinetically constrained models (KCMs) were introduced by physicists to model the liquid-glass transition. They are interacting particle systems on in which each element of can be in state 0 or 1 and tries to update its state to 0 at rate and to 1 at rate , provided that a constraint is satisfied. In this article, we prove the first non-perturbative result of convergence to equilibrium for KCMs with general constraints: for any KCM in the class termed “supercritical” in dimension 1 and 2, when the initial configuration has product law with , the dynamics converges to equilibrium with exponential speed when is close enough to 1, which corresponds to the high temperature regime.
2010 Mathematics Subject Classification: 60K35.
**Key words: Interacting particle systems; Glauber dynamics; kinetically constrained models; bootstrap percolation; convergence to equilibrium. **
1. Introduction
Kinetically constrained models (KCMs) are interacting particle systems on , in which each element (or site) of can be in state 0 or 1. Each site tries to update its state to 0 at rate and to 1 at rate , with fixed, but an update is accepted if and only if a constraint is satisfied. This constraint is defined via an update family , where and the , called update rules, are finite nonempty subsets of : the constraint is satisfied at a site if and only if there exists such that all the sites in have state zero. Since the constraint at a site does not depend on the state of the site, it can be easily checked that the product measure, , satisfies the detailed balance with respect to the dynamics, hence is reversible and invariant. is the equilibrium measure of the dynamics.
KCMs were introduced in the physics literature by Fredrickson and Andersen [11] to model the liquid-glass transition, an important open problem in condensed matter physics (see [22, 13]). In addition to this physical interest, KCMs are also mathematically challenging, because the presence of the constraints make them very different from classical Glauber dynamics and prevents the use of most of the usual tools.
One of the most important features of KCMs is the existence of blocked configurations. These blocked configurations imply that the equilibrium measure is not the only invariant measure, which complicates a lot the study of the out-of equilibrium behavior of KCMs; even the basic question of their convergence to remains open in most cases.
Because of the blocked configurations, one cannot expect such a convergence to equilibrium for all initial laws. Initial measures particularly relevant for physicists are the with (see [17]). Indeed, is a measure of the temperature of the system: the closer is to 0, the lower the temperature is. Therefore, starting the dynamics with a configuration of law means starting with a temperature different from the equilibrium temperature. In this case, KCMs are expected to converge to equilibrium with exponential speed as soon as no site is blocked for the dynamics in a configuration of law or . However, there have been few results in this direction so far (see [7, 3, 8, 21, 18]), and they have been restricted to particular update families or initial laws.
Furthermore, general update families have attracted a lot of attention in recent years. Indeed, there recently was a breakthrough in the study of a monotone deterministic counterpart of KCMs called bootstrap percolation. Bootstrap percolation is a discrete-time dynamics in which each site of can be infected or not; infected sites are the bootstrap percolation equivalent of sites at zero. To define it, we fix an update family and choose a set of initially infected sites; then for any , the set of sites that are infected at time is
[TABLE]
which means that the sites that were infected at time remain infected at time and a site that was not infected at time becomes infected at time if and only if there exists such that all sites of are infected at time . Until recently, bootstrap percolation had only been considered with particular update families, but the study of general update families was opened by Bollobás, Smith and Uzzell in [6]. Along with Balister, Bollobás, Przykucki and Smith [1], they proved that general update families satisfy the following universality result: in dimension 2, they can be sorted into three classes, supercritical, critical and subcritical (see definition 2), which display different behaviors (their result for the critical class was later refined by Bollobás, Duminil-Copin, Morris and Smith in [5]).
These works opened the study of KCMs with general update families. In [20, 19, 14, 15], Hartarsky, Martinelli, Morris, Toninelli and the author showed that the grouping of two-dimensional update families into supercritical, critical and subcritical is still relevant for KCMs, and established an even more precise classification. However, these results deal only with equilibrium dynamics. Until now, nothing had been shown on out-of-equilibrium KCMs with general update families, apart from a perturbative result in dimension 1 [7].
In this article, we prove that for all supercritical update families, for any initial law , , when is close enough to 1, the dynamics of the KCM converges to equilibrium with exponential speed. This result holds in dimension 2 and also in dimension 1 for a good definition of one-dimensional supercritical update families. It is the first non-perturbative result of convergence to equilibrium holding for a whole class of update families.
This result may help to gain a better understanding of the out-of-equilibrium behavior of supercritical KCMs. In particular, such results of convergence to equilibrium were key in proving “shape theorems” for specific one-dimensional constraints in [2, 12, 4].
2. Notations and result
Let . We denote by the -norm on . For any set , will denote the cardinal of .
For any configuration , for any , we denote the value of at . Moreover, for any , we denote the restriction of to , and (or just 0 when is clear from the context) the configuration on that contains only zeroes.
We set an update family with and the finite nonempty subsets of . To describe the classification of update families, we need the concept of stable directions.
Definition 1**.**
For , we denote the half-space with boundary orthogonal to . We say that is a stable direction for the update family if there does not exist such that ; otherwise is unstable. We denote by the set of stable directions.
[6] gave a classification of two-dimensional update families into supercritical, critical or subcritical depending on their stable directions. Here is the generalization proposed for -dimensional update families by [5] (definition 9.1 therein), where for any , is the interior of in the usual topology on .
Definition 2**.**
A -dimensional update family is
- •
supercritical if there exists an open hemisphere that contains no stable direction;
- •
critical if every open hemisphere contains a stable direction, but there exists a hemisphere such that ;
- •
subcritical if for every hemisphere .
Our result will be valid for supercritical update families.
The KCM process with update family can be constructed as follows. We set . Independently for all , we define two independent Poisson point processes and on , with respective rates and . We call the elements of clock rings and denote them by . The elements of will be 0-clock rings and the elements of will be 1-clock rings. For any intial configuration , we construct the KCM as the continuous-time process on defined thus: for any , for , and for any ,
- •
if there exists such that , then for , where is a -clock ring, ;
- •
if such an does not exist, for .
In other words, sites try to update themselves to 0 when there is a 0-clock ring, which happens at rate , and to 1 when there is a 1-clock ring, which happens at rate , but an update at is successful if and only if there exists an update rule such that all sites of are at zero. This construction is known as Harris graphical construction. One can use the arguments in part 4.3 of [24] to see that it is well-defined. We denote by the law of when the initial configuration has law .
For any , we denote the product measure. Since the constraint at a site does not depend on the state of the site, it can be easily checked that satisfies the detailed balance with respect to the dynamics, hence is reversible and invariant. is called equilibrium measure of the dynamics.
We will say that a function is local if its output depends only on the states of a finite set of sites, and we then denote its norm.
Theorem 3**.**
If or 2, for any supercritical update family , for any , there exists such that for any , for any local function , there exist two constants and such that for any ,
[TABLE]
Remark 4*.*
We expect theorem 3 to hold also for . However, our proof relies on proposition 13, which is easy for and was proven in [6] for , but for which there is no equivalent for . Such an equivalent would extend our result to .
The remainder of this article is devoted to the proof of theorem 3. The argument is based on the proof given in [21] for the particular case of the Fredrickson-Andersen one-spin facilitated model, but brings in novel ideas in order to accommodate the much greater complexity of general supercritical models. From now on, we fix or 2 and a supercritical update family in dimension . We begin in section 3 by using the notion of dual paths to reduce the proof of theorem 3 to the simpler proof of proposition 7. Then in section 4 we use the concept of codings to simplify it further, reducing it to the proof of proposition 11. In section 5 we introduce an auxiliary oriented percolation process, that we use in section 6 to prove proposition 11 hence theorem 3.
3. Dual paths
In this section, we use the concept of dual paths to reduce the proof of theorem 3 to the easier proof of proposition 7. Let . We notice that the Harris graphical construction allows us to couple a process with initial law and a process with initial law by using the same clock rings but different initial configurations (independent of the clock rings and of each other). We denote the joint law by . We notice that since is an invariant measure for the dynamics, has law for all . To prove theorem 3, it is actually enough to show
Proposition 5**.**
For any , there exists such that for any , there exist two constants and such that for any and , .
Indeed, if is a local function depending of a finite set of sites ,
[TABLE]
[TABLE]
Therefore we will work on proving proposition 5.
In order to do that, we need to introduce dual paths. We define the range of by
[TABLE]
For any , and , a dual path of length starting at (see figure 1) is a right-continuous path that starts at site at time , goes backwards, is allowed to jump only when there is a clock ring, and only to a site within -distance . To write it rigorously, the path satisfies and there exists a sequence of times satisfying the following properties: for all and all , , and for all , and .
We denote the (random) set of all dual paths of length starting from . A dual path is called an activated path if it “encounters a point at which both processes are at 0”, i.e. if there exists such that . The set of all activated paths in is called . We have the
Lemma 6**.**
For any and , if , then for all , .
Sketch of proof..
The proof is the same as for lemma 1 of [21], apart from the fact that if the path is at , it does not necessarily jump to a neighbor of , but to an element of , . The idea of the proof is to start a dual path at , where the two processes disagree, and, staying at , to go backwards in time until the processes agree at . At this time, there was an update at in one process but not in the other, hence an update rule that was full of zeroes in one process but not in the other, thus a site at distance at most of at which the two processes disagree. We jump to this site and continue to go backwards. This construction yields a dual path along which the two processes disagree, hence they can not be both at zero, so the path is not activated. ∎
Lemma 6 implies that to prove proposition 5 hence theorem 3, it is enough to prove
Proposition 7**.**
For any , there exists such that for any , there exist two constants and such that for any , , there exists such that .
The remainder of the article will be devoted to the proof of proposition 7.
4. Codings
This section is devoted to the reduction of the proof of proposition 7 (hence of theorem 3) to the simpler proof of proposition 11, via the use of codings. The idea is the following: in order to prove proposition 7, it is enough to show that along each dual path, the two processes are at zero at one of the discrete times [math], , , etc. hence we only need to consider the positions of the path at these times, which will make up the coding of the path. Let and . A coding is a sequence of sites in . Moreover, for and , the coding of is the sequence . If is a coding, we define the event . If is satisfied, is an activated path.
Therefore, to prove proposition 7 hence theorem 3, it is enough to prove
Proposition 8**.**
For any , there exists such that for any , there exist two constants and and a constant such that for any and , .
Proposition 8 holds only for greater than a constant, but this is enough, since we only have to enlarge to obtain a bound valid for all .
In order to prove proposition 8, we will define a set of “reasonable codings” and prove that the probability that there exists a dual path whose coding is not in decays exponentially in (lemma 9). Then we will count the number of codings in (lemma 10). Therefore it will be enough to give a bound on for any to prove proposition 8 hence theorem 3. Such a bound is stated in proposition 11 and will be proven in section 6.
For any constant , for any , and , the set of “reasonable codings” is defined as the set of where is a sequence of sites satisfiying , and for all and where satisfy . We can now state lemmas 9 and 10, as well as proposition 11. These statements together prove proposition 8.
Lemma 9**.**
For any , there exists such that for any , , there exists a constant such that for all and , .
In the following, will always be the given by lemma 9.
Lemma 10**.**
There exist constants and such that for any , and , .
Proposition 11**.**
For any , there exists a constant such that for any , there exists such that for any , there exist two constants and such that for any , and , .
We are now going to prove lemmas 9 and 10. After that, it will suffice to prove proposition 11 to prove theorem 3.
Sketch of proof of lemma 9..
This can be proven with the argument of the lemma 5 of [21]; the idea is that if there exists with , there are so many clock rings that the probability becomes very small. Indeed, let us say visits the sites in the time interval , then the sites in the time interval , etc. until the sites in the time interval . Then the coding of is , hence implies . It yields that visits more than sites in a time , and there must be successive clock rings at these sites. The proof of lemma 5 of [21] yields that we can choose large enough depending on , hence on , such that the probability of this event is at most with . ∎
To prove lemma 10, we need the following classical combinatorial result, which will also be used in the proof of lemma 19.
Lemma 12**.**
For any , . Moreover, for any , .
The proof of the first part of lemma 12 can be found just before the section 2 of [16] and the proof of the second part in section 1.2 of [23] (weak compositions).
Proof of lemma 10..
Let , and . By definition, elements of have the form with satisfiying , and for all , and with satisfying . Therefore, to count the number of elements of , it is enough to count the number of possible and the number of possible given .
We begin by counting the number of possible . We have . Moreover, by the second part of lemma 12, for any integer , the number of possible sequences of integers such that is at most , hence the number of possible is at most by the first part of lemma 12. Furthermore by the Stirling formula, where is a constant. In addition, since , , hence the number of possible is at most as and is large.
We now fix a sequence and count the possible . We know that . Moreover, for all , , hence for each , we have , so there are at most choices for given . Therefore the number of choices for is at most . Moreover, for any and any positive , we have , therefore the number of choices is bounded by
[TABLE]
since . As , , thus the number of choices for given is bounded by . ∎
5. An auxiliary process
In order to prove proposition 11, we need to find a mechanism for the zeroes to spread in the KCM process; this mechanism uses novel ideas to deal with the complexity of general supercritical models. We begin in section 5.1 by using the bootstrap percolation results of [6] to find a mechanism allowing the zeroes to spread locally (proposition 13). Then we use it in section 5.2 to define an auxiliary oriented percolation process which guarantees that if certain conditions are met, the KCM process is at zero at a given time (proposition 15). Finally, in section 5.3 we prove some properties of this auxiliary process that we will use in section 6.
5.1. Local spread of zeroes
This is the place where we need the supercriticality of . Indeed, since is supercritical, the results of [6] yield the following proposition (see figure 2):
Proposition 13** ([6]).**
For or , there exists , a rectangle of the following form:
- •
if , with ;
- •
if , with , where is a vector orthogonal to ,
and a sequence of sites in such that if the sites of are at zero and there are successive 0-clock rings at while there is no 1-clock ring in , the sites of are at zero afterwards.
Remark 14*.*
For , we expect a similar proposition to hold, maybe with , contained in the hyperplane orthogonal to , but we can not prove it because an equivalent of the construction of [6] is not available yet. Proving such a construction would be enough to extend our result to any dimension.
Proof of proposition 13..
We begin with the case . Since is supercritical there exists an unstable direction. Without loss of generality we can say that , therefore there exists an update rule contained in . This yields the mechanism illustrated by figure 3(a): if is sufficiently large and full of zeroes, is full of zeroes, hence if the site receives a 0-clock ring, this clock ring puts it at zero. Then is full of zeroes, thus if receives a 0-clock ring, this clock ring puts it at zero. In the same way, if the sites receive successive 0-clock rings, these clock rings will put them successively at zero, therefore will be at zero. This yields the result with and .
We now consider the case . Since is supercritical, there exists a semicircle in that contains no stable direction; we call its middle. The results of section 5 of [6] (see in particular figure 5 and lemma 5.5 therein) prove that there exists a set of sites, called a droplet, such that in the bootstrap percolation dynamics, if we start with all the sites of the droplet infected, other sites in the direction can be infected, creating a bigger infected droplet with the same shape (see figure 3(b)). We can enlarge this droplet into a rectangle as in figure 3(c); furthermore can be chosen rational111Indeed, theorem 1.10 of [6] states that the set of stable directions is a finite union of closed intervals with rational endpoints, hence the semicircle containing no stable direction can be chosen with rational endpoints., hence we may enlarge enough so that . Now, since contains the original droplet, if is infected the infection can grow from the droplet into a droplet big enough to contain while staying in (see figure 3(c)). We call the sites that are successively infected during this growth (sites infected at the same time are ordered arbitrarily). Since is the first site infected by the bootstrap percolation dynamics starting with the sites of infected, there exists an update rule such that , therefore if the KCM dynamics starts with all the sites of at zero and there is a 0-clock ring at , this clock ring sets to zero. Then, if there is a 0-clock ring at , it will set to zero for the same reason, and successive 0-clock rings at will set them successively to 0, which puts at zero.
∎
5.2. Definition of the auxiliary process
Let , and . For any and , we will define an oriented percolation process on , from time zero to time (see [9] for an introduction to oriented percolation). For and with even, the bonds and can be open or closed. We set , and for any , with even, if and only if and the bond is open or and the bond is open. For any , with odd, we set .
The state of the bonds is defined as follows. For any , with even:
- •
is open if and only if
[TABLE]
i.e. there is no 1-clock ring in during the time interval ;
- •
is open if and only if
[TABLE]
i.e. there are successive 0-clock rings in the equivalent of for during the time interval , and no 1-clock ring at these sites or in in this time interval.
We notice that if all the sites of are at zero at time and is open, the sites of are still at zero at time . Moreover, by proposition 13, if the sites of are at zero at time and is open, the sites of are at zero at time . This allows us to deduce (see figure 4 for an illustration of the mechanism):
Proposition 15**.**
If there exists such that and the sites of are at zero at time , then the sites of are at zero at time .
5.3. Properties of the auxiliary process
In this subsection we state the two oriented percolation properties of , propositions 17 and 18, that we will use in section 6 to prove proposition 11. In order to do that, we need a bound on the probability that a bond is closed; this will be lemma 16. It is there that we need bigger than a ; this is necessary so that the probability that there is no 1-clock ring at the sites we consider is large. For any , we set . We can then state
Lemma 16**.**
There exists a constant such that for , , , and , the probability that any given bond is closed for the process is smaller than .
Proof.
Let , , , and . Let , with even. We notice that if the bond is closed, the bond is also closed, hence it is enough to bound the probability that is closed. Denoting and , we need to bound the probabilities of and . We begin with . The events are independent and have probability each; moreover, belong to , so ; we deduce the probability of is
[TABLE]
thus the probability of is at most . Moreover, the probability of is the probability that a Poisson point process of parameter has strictly less than elements in an interval of length , hence it is . When is large enough, , hence this probability is smaller than , which is smaller than when is large enough depending on , hence on . Consequently, when is large enough depending on , the probability that is closed is smaller than , which is smaller than . ∎
Thanks to lemma 16, it is possible to prove two oriented percolation properties of . Firstly, for any , , , and , we define the time of death of the process (if the set is empty, is infinite). Since , which is not identically zero, . Then we have
Proposition 17**.**
For any , there exists a constant such that for any , , , , , , .
Sketch of proof..
The proposition can be proven by a classical contour method like the one presented in section 10 of [9]. The idea is that if we can draw a “contour of closed bonds” around the connected component of ones in , and this contour will have length . Furthermore, it can be seen that bonds separated by at least 5 bonds from each other are independent, because they depend on clock rings in disjoint space-time intervals. Therefore if we keep one bond out of 6, we extract independent closed bonds from the contour, each of them having probability from lemma 16 when , hence the bound. ∎
also satisfies a second property. For any , , , and , we denote . Then we have
Proposition 18**.**
For any , , there exists a constant such that for any , there exist constants and such that for any , , and , .
Sketch of proof..
This proposition comes from classical results in oriented percolation. Firstly, if the process survives until time , it has a big “range”, which means that if we define and , and are so large ; this can be proven with the contour argument in section 11 of [9]. Moreover, the argument that proves (1) in [9] also proves that in , coincides with the oriented percolation process that has the same bonds, but which starts with all sites at 1 instead of just the origin. Finally, the end of section 5 of [10] contains a contour argument for the latter process which allows to prove that it has a lot of ones; we can use this argument with the same adaptations we used for the contours of proposition 17. ∎
6. Proof of proposition 11
In this section we use the auxiliary process defined in section 5 to give a proof of proposition 11. In order to do that, we need some definitions. For any , , , , and , we define if such a exists; in this case we also denote (in the following, when we write or without more precision, we always assume that they exist). For any we define the events
[TABLE]
By proposition 15, if , then the sites of are at zero at time in both processes and , in particular is at zero at time in both processes, therefore is satisfied. Consequently,
[TABLE]
Therefore we only have to prove the following lemmas 19, 20 and 21 to prove proposition 11, thus ending the proof of theorem 3:
Lemma 19**.**
For any , there exists a constant such that for any , , there exist constants and such that for any , , , we have does not exist.
Lemma 20**.**
For any , there exists a constant such that for any , , there exist constants and such that for any , , , exists, .
Lemma 21**.**
For any , , , there exists a constant such that for any , , , we get and .
Proof of lemma 19..
We set , which depends only on . Let , , , , and . If does not exist, is finite for , therefore if we call and for , is finite as long as . We will use proposition 17 to bound the probability that this happens. We call ; we then have , hence if is the integer satisfying , we have . Furthermore, if are integers satisfiying for , we get , for all (we denote ). In addition, since for any , . We deduce
[TABLE]
[TABLE]
Moreover, the events , and depend only on clock rings in the time intervals , and , which are disjoint, thus the events are independent, hence
[TABLE]
[TABLE]
[TABLE]
by proposition 17 and since . Consequently,
[TABLE]
In addition, lemma 12 yields that for any , we have , and by the Stirling formula there exists a constant such that
[TABLE]
since . We deduce . Therefore
[TABLE]
[TABLE]
In addition, since , , so , hence
[TABLE]
which is the lemma. ∎
Proof of lemma 20..
This proof is an application of proposition 18. We set , which depends only on . Let , , and . It is enough to prove the lemma for ; indeed, if the lemma holds for , one has only to enlarge to prove it for . Therefore we set and . If exists but , we have and , hence
[TABLE]
We are going to bound the term on the right. For any , we have , and since , thus , so , hence if we choose we have . Therefore by proposition 18,
[TABLE]
since . Consequently
[TABLE]
which yields lemma 20. ∎
Proof of lemma 21..
Let , , , , and . The argument is elementary: we notice that there is a positive probability that a rectangle is full of zeroes in the initial configurations of the two processes since they have laws and , as well as a positive probability that there is no 1-clock ring in the rectangle in the time interval . Therefore there is a positive probability that a rectangle is full of zeroes in both processes at time , so if there are elements in , the probability that none of the corresponding rectangles is full of zeroes in both processes at time is of order .
We notice that depends only on clock rings in the time interval , hence if is the -algebra generated by the clock rings in , for or , we have
[TABLE]
Moreover,
[TABLE]
[TABLE]
[TABLE]
[TABLE]
since the events or depend only on the state of and on the clock rings of the time interval at the sites of , so they are mutually independent and independent of . Therefore the invariance by translation yields
[TABLE]
[TABLE]
[TABLE]
Furthermore, since and ,
[TABLE]
since . This implies
[TABLE]
In addition, if , , so , and if , . Moreover, since , , hence . This implies that if is the minimum of and (which depends only on and ), for or we have . Consequently, (1) yields
[TABLE]
[TABLE]
which is the lemma. ∎
Acknowledgements
I would like to thank my PhD advisor Cristina Toninelli; I also would like to thank Ivailo Hartarsky for his careful reading of this paper and for his suggestions, as well as for pointing me to some references.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Paul Balister, Béla Bollobás, Michał Przykucki, and Paul Smith. Subcritical 𝒰 𝒰 \mathcal{U} -bootstrap percolation models have non-trivial phase transitions. Transactions of the American Mathematical Society , 386:7385–7411, January 2016.
- 2[2] Oriane Blondel. Front progression in the East model. Stochastic processes and their applications , 123(9):3430–3465, September 2013.
- 3[3] Oriane Blondel, Nicoletta Cancrini, Fabio Martinelli, Cyril Roberto, and Cristina Toninelli. Fredrickson-Andersen one spin facilitated model out of equilibrium. Markov processes and related fields , 19(3):383–406, 2013.
- 4[4] Oriane Blondel, Aurélia Deshayes, and Cristina Toninelli. Front evolution of the Fredrickson-Andersen one spin facilitated model. Electronic journal of probability , 24:32, January 2019.
- 5[5] Béla Bollobás, Hugo Duminil-Copin, Robert Morris, and Paul Smith. Universality of two-dimensional critical cellular automata. Proceedings of the London Mathematical Society , to appear.
- 6[6] Béla Bollobás, Paul Smith, and Andrew Uzzell. Monotone cellular automata in a random environment. Combinatorics, probability and computing , 24(4):687–722, July 2015.
- 7[7] Nicoletta Cancrini, Fabio Martinelli, Roberto H. Schonmann, and Cristina Toninelli. Facilitated oriented spin models: some non equilibrium results. Journal of statistical physics , 138(6):1109–1123, 2010.
- 8[8] Paul Chleboun, Alessandra Faggionato, and Fabio Martinelli. Mixing time and local exponential ergodicity of the East-like process in ℤ d superscript ℤ 𝑑 \mathds{Z}^{d} . Annales de la faculté des sciences de Toulouse , 24(4):717–743, 2015.