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FA-1f and other non-cooperative models
Apart from the East model, the most famous KCSM is the Fredrickson-Andersen model FA-1f (for “Fredrickson-Andersen one-spin facilitated model”). In that case the constraint is satisfied at x iff x has at least one vacant nearest neighbour. Put another way, for the FA-1f model cx() = 1 Y yx y: (1.5).
This model has more symmetries than the East model: it is invariant by permutation of the base vectors fe1;e1; : : : ; ed;edg. It is also non-cooperative, in the sense that it is enough to have a fixed finite set of zeros in the initial configuration to be able to empty any site in Zd using only flips authorized by the constraints. For FA-1f in fact, one zero is enough. More formally, one can give the following definition of non-cooperative models. Definition 1.1.6 A KCSM is non-cooperative if there exists a finite set S of Zd with the following property: if y = 0 for all y 2 S, then for any x 2 Zd there exists a finite sequence (0); : : : ; (n) 2 such that (0) = , (n) x = 0 and (i+1) = (i) xi for some xi 2 Zd where cxi (i) = 1. A set S satisfying the previous condition is called a seed. Models that are not non-cooperative are called cooperative.
The name “non-cooperative” comes from the fact that in such a model there is no need for an infinite number of zeros to cooperate to be able to empty any arbitrary site. On the contrary, the East model is cooperative: if there are only finitely many zeros in the initial configuration, no site on the right of the rightmost zero can ever have its constraint satisfied. The generalization of the FA-1f model, FA-jf, for which the constraint requires at least j empty nearest neighbours, is also cooperative for j = 2; : : : ; 2d. Note that the seed S in the definition is not unique; in particular, since the constraints are translation invariant, any translation of S is again a seed.
Ergodicity and equilibrium relaxation
In Chapter 2, I will come back to the question of relaxation when the system starts out of equilibrium, which can correspond for instance to a rapid cool-down (Section 2.1) or a closer investigation of the structure of the dynamics (Section 2.2). In Chapter 3, I review what can be said about characteristic times (relaxation time, persistence time and diffusion coefficient) at low temperature, i.e. when q ! 0. Dynamical heterogeneity, characterized by the existence of active and less active regions which increase in size as q ! 0, will be a transversal point of interest during these investigations. Before turning to these issues, let us see how to identify the ergodicity regime of KCSMs and what can be said about equilibrium relaxation.
Bootstrap percolation and ergodicity
Due to the presence of degenerate transition rates, classic arguments to show ergodicity of the process ([Lig85]) do not apply to KCSMs. In [CMRT08], the authors identify the ergodicity region of KCSMs and characterize it in terms of sub-critical regime of a certain bootstrap percolation, associated to the specific constraints of the model. Let us first define this percolation. Fix a KCSM with constraints (cx())2 ;x2Z and define the bootstrap map B : ! by B()x = 0 if x = 0 or cx() = 1 1 else. (1.8).
In words, the effect of this application is to empty the sites everywhere it is allowed by the constraints. If we apply it infinitely many times starting from , we get a limit distribution on . The criterium for ergodicity is then whether or not this limit distribution is 0, the distribution that charges only the empty configuration. Proposition 1.3.1 [CMRT08] The KCSM with constraints (cx())2 ;x2Z at density p is ergodic iff, starting from the Bernoulli(p) product measure, almost surely the origin is emptied after finitely many iteration of the bootstrap procedure. Here ergodic means for instance that if f 2 L2() is invariant for the equilibrium dynamics (in the sense that for all t > 0 E [f((t))] = f() d()-a.s.), then it is constant -a.s., i.e. 0 is a simple eigenvalue of L. Equivalently, V ar (E [f((t))]) ! t!1 0 8f 2 L2().
Local relaxation in the FA-1f model
In the following paragraph, we will see that for the East model we can answer the question of the out-of-equilibrium relaxation in an almost optimal way with the help of the distinguished zero. The models in which one can define an analogue of the distinguished zero are the only ones for which available results are so complete. The only other class of models for which we can give a partial answer is that of non-cooperative models. This is a joint work with Nicoletta Cancrini, Fabio Martinelli, Cyril Roberto and Cristina Toninelli ([BCM+13]), which has been accepted for publication in Markov Processes and Related Fields and is presented in Appendix A. The main result is the following (Theorem A.2.1), stated here in lesser generality (only on Zd, while the techniques allow to treat any graph with polynomial growth). The proof could also be adapted to more general non-cooperative models.
Asymptotics for the spectral gap at low temperature
As the temperature goes to 0, so does the spectral gap. Equilibrium relaxation still takes place with exponential decay but the time scales involved diverge because zeros become fewer and thus it is increasingly costly to satisfy a constraint. A simple test function proving this fact is f() = 0, which verifies D(f)=V ar(f) = (c0), where c0 is the constraint at 0. Therefore the variational definition of gap (1.10) implies that gap 6 (c0), which goes to zero as soon as the constraint c0 requires at least a zero in a finite neighbourhood of the origin. This however is a poor lower bound for the divergence of the relaxation time. Indeed, mechanisms more complex than the mere rarefaction of zeros are involved in this divergence, which I explain in the next paragraphs in the cases of the East and non-cooperative models.
Table of contents :
Remerciements
Résumé détaillé
1 Introduction
1.1 Definition and first properties of KCSMs
1.1.1 General description of KCSMs
1.1.2 The East model and its distinguished zero
1.1.3 FA-1f and other non-cooperative models
1.2 Physical motivations
1.3 Ergodicity and equilibrium relaxation
1.3.1 Bootstrap percolation and ergodicity
1.3.2 Spectral gap
2 Out-of-equilibrium dynamics
2.1 Out-of-equilibrium relaxation
2.1.1 Preliminary remarks
2.1.2 Local relaxation in the FA-1f model
2.1.3 Relaxation in the East model
2.2 Bubbles and front
3 Low temperature dynamics
3.1 Asymptotics for the spectral gap at low temperature
3.1.1 Asymptotics for East and energy barriers
3.1.2 Asymptotics for FA-1f and conjectures for non-cooperative models .
3.2 Diffusion coefficient and Stokes-Einstein relation
A Out of equilibrium relaxation in the FA-1f model
A.1 Introduction
A.2 Notation and Result
A.2.1 The graph
A.2.2 The probability space
A.2.3 The Markov process
A.2.4 Main Result
A.3 A preliminary result on Markov processes
A.4 Persistence of zeros out of equilibrium
A.5 Proof of the main theorem
A.6 Spectral gap on the ergodic component
B Front progression in the East model
B.1 Introduction
B.2 Model
B.2.1 Setting and notations
B.2.2 Former useful results
B.3 Preliminary results
B.4 Decorrelation behind the front
B.4.1 Presence of voids behind the front
B.4.2 Relaxation to equilibrium on the left of a distinguished zero
B.4.3 Decorrelation behind the front at finite distance
B.5 Invariant measure behind the front
B.6 Front speed
C Tracer diffusion in low temperature KCSM
C.1 Introduction
C.2 Models and notations
C.3 Convergence to a non-degenerate Brownian motion
C.4 Asymptotics for D in non-cooperative models
C.4.1 Lower bound in Theorem C.4.1
C.4.2 Upper bound in Theorem C.4.1
C.5 In the East model, D gap
C.6 An alternative proof in the FA-1f model
C.7 Lower bound for the windmill model
D Stokes-Einstein breakdown in KCSM?
