Asymptotics for critical kinetically constrained models with an infinite number of stable directions

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Advances of the thesis and universality partition

In this subsection we present the new results proven in this thesis, as well as the universality classification of KCMs they allow to deduce. They show that though the universality classes observed in bootstrap percolation are relevant and give rise to different behaviors in KCMs, they are not precise enough, because KCMs in the same class may behave differently. Therefore a refinement of this classification was needed for KCMs; the work of this thesis was key in proving this refinement, which is presented below.
We begin by considering supercritical update families. In order to present the refinement of the universality partition introduced for KCMs, we need the following definition for the subclasses:
Definition 2.10. A two-dimensional supercritical update family is:
rooted if it has two non opposite stable directions;
unrooted otherwise.
One can see that the 2-dimensional East model is rooted and that the FA-1f model is unrooted (their stable directions are featured on figure 1). The following result is proven in this thesis:
Theorem 1. For any two-dimensional supercritical rooted update family, Eq (KCM) = 1=q (ln(1=q)) when q tends to 0, and the same holds for Trel.
Theorem 1 relies on a combinatorial result (theorem 7) proven in the article [Mar17], which corresponds to the chapter 1 of this thesis. Theorem 1 itself was proven in collaboration with Fabio Martinelli and Cristina Toninelli in the article [MMTar], which corresponds to the chapter 2 of this thesis.
Remark 2.11. The proof of theorem 1 works in a more general setting, for all two-dimensional update families that are not supercritical unrooted, but the theorem is sharp only for supercritical rooted update families. Contrary to the other universality results proven so far for bootstrap percolation and KCMs, theorem 1 also holds in any dimension with the following generalization of the definition of a supercritical unrooted update family: a d-dimensional update family is supercritical unrooted if there exists a hyperplane of Rd containing all the stable directions of U, stable directions being defined in the same way as in dimension 2.
Combining theorem 1 with the upper bounds proven in [MMT19] by Martinelli, Morris and Toninelli (and with the easy lower bounds for supercritical unrooted models mentioned in remark 2.8) allows to prove the following universality result for supercritical KCMs:
Theorem 2. For any two-dimensional supercritical update family U.
if U is unrooted, Eq (KCM) = 1=q(1) when q tends to 0.
if U is rooted, Eq (KCM) = 1=q(ln(1=q)) when q tends to 0.

Sketch of the proof of proposition 1.6

For d = 1, if U is a supercritical unrooted family, it has no stable direction, therefore there must be an update rule contained in N and another contained in 􀀀N. Consequently, as illustrated by figure 1.5, if we have an interval I Z of zeroes that is sufficiently large, the site s at the right of I can be put at zero with a legal move. Then the site s0 at the left of the interval can be put at one by a legal move, and I has moved to the right by one unit. By having I starting from outside the domain (where there are only zeroes) and moving towards the origin in that way, one can put the origin at zero using a bounded number of zeroes, whatever the size of the domain.
For d = 2 the mechanism is similar, but requires a more complex construction. In section 5 of [BSU15] (see in particular figure 5 and lemma 5.5 therein), it is proven that if U is an update family with a semicircle of unstable directions centered on direction u, it is possible to construct a “droplet”: a finite set of zeroes that even if all other sites are at 1, allows us to put more sites at zero in direction u with legal moves, creating a bigger droplet of the same shape, as illustrated on part (a) of figure 1.6. It is the shape of the part of the droplet towards direction u that enables its growth towards this direction. If U is supercritical unrooted, its stable directions are contained in a hyperplane of R2, which means a straight line, hence there are at most two stable directions, and they must then be opposite. Therefore, there exist two opposite semicircles containing no stable direction, with middles u and 􀀀u.
We can use the construction of [BSU15] to build two droplets, corresponding to the two semicircles, that can grow respectively in the directions u and 􀀀u (see part (b) of figure 1.6). Using these two droplets, we can get a combined droplet that can grow in both directions u and 􀀀u (part (c) of figure 1.6).

Supercritical rooted KCMs

In order to define the class of supercritical rooted update families we should begin by recalling the key geometrical notion of stable directions introduced in [BSU15]. Given a unit vector u 2 S1, let Hu := fx 2 Z2 : hx; ui < 0g denote the discrete half-plane whose boundary is perpendicular to u. Then, for a given update family U, the set of stable directions is S = S(U) = u 2 S1 : [Hu] = Hu  : The update family U is supercritical if there exists an open semicircle in S1 that is disjoint from S. In [BSU15] it was proven that for each supercritical update family the median of the infection time of the U-bootstrap process diverges as 1=q(1). In [Mor17a], the author R. Morris, together with two of us, conjectured that not all supercritical update families give rise to the same scaling for KCMs and that the supercritical class should be refined into two subclasses to capture the KCMs scaling as follows.

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East-like motion of the arrows and proof of proposition

Let A` = f! 2: !I+ 0 0g [ f! 2 : !I􀀀 0 0g; where I 0 = f(0;1); : : : ; (0;`)g. Then it holds Lemma 2.29. If A »;q \ A 6= ; then there exist ! 2 A` and a legal path connecting g to ! such that \ Bi(ni) = ;; i = 1; 2.
Proof. Fix ! 2 A »;q \ A, recall definition 2.25 and let ~ be a legal path connecting g to (!V ; ~!V c 0) such that ~ \B1(n1􀀀1) = ; and ~ \B2(n2􀀀 1) = ;. Without loss of generality, we can assume that ~ ends as soon as the origin is infected. It is easy to verify that ~ must be able to sequentially infect (and possibly heal later on) the ordered vertices of either I+ 0 starting from (0; `) or those of I􀀀0 starting from (0;􀀀`). For simplicity we assume that the first option holds and we let be the path obtained from ~ by deleting all the transitions in which a vertex of I+ 0 is healed.
By construction, the final configuration of belongs to A`. Moreover, is a legal path because at each step the infection in the last column of V is larger than or equal to the infection of the corresponding step of ~ . Finally the restriction to C1; : : : ; CN􀀀1 of any step of coincides with the same restriction of the appropriate step of ~ . Using that ~ \ B1(n1 􀀀 1) = ; and ~ \B2(n2􀀀1) = ;, we deduce that \B1(n1) = ; and \B2(n2) = ;. The above lemma says that, if there exists a configuration in g for which we can infect the origin performing a legal path never crossing either B1(n1􀀀 1) or B2(n2􀀀1), then necessarily there exists a legal path never crossing either B1(n1) or B2(n2) and connecting a configuration ! with all columns being # to a configuration ! with a  » in the N-th column. In order to conclude that A »;q \ A = ; and thus prove our proposition 2.26, we will now show that the existence of a legal path with the above properties is impossible. It is here that the East-like motion of the droplets emerges and plays a key role. Recall the definitions (2.9), (2.15) and let m = 4n1n2 and, for simplicity, let us suppose that m divides N. We partition [N] into M = N=m disjoint consecutive blocks fBigMi =1 of equal cardinality and, with a slight abuse of notation, we identify the columns [k2BiCk with the block Bi itself. Given ! 2 we writi(!) := 1f8 j 2Bi : (!)j=#g.

Table of contents :

Résumé détaillé
Introduction
1 Definitions and tools
1.1 Definitions
1.2 Bootstrap percolation
1.3 Relaxation time
2 Universality
2.1 Previous results
2.2 Advances of the thesis and universality partition
2.3 Sketch of proofs
3 Convergence to equilibrium
3.1 Previous results
3.2 Advances of the thesis
3.3 Sketch of proofs
1 Combinatorics for supercritical rooted kinetically constrained models 
1.1 Introduction
1.2 Notations and result
1.3 The one-dimensional case
1.4 The general case
1.5 Sketch of the proof of proposition 1.6
2 Asymptotics for Duarte and supercritical rooted kinetically constrained models 
2.1 Introduction
2.2 Models and notation
2.2.1 Notation
2.2.2 Models
2.3 A variational lower bound for E(0)
2.4 Supercritical rooted KCMs
2.5 The Duarte KCM
2.5.1 Preliminary tools: the Duarte bootstrap process
2.5.2 Algorithmic construction of the test function and proof of theorem 2.11
2.5.3 East-like motion of the arrows and proof of proposition
2.5.4 Density of droplets and proof of proposition 2.27
2.5.5 Finishing the proof of proposition 2.27
3 Asymptotics for critical kinetically constrained models with an infinite number of stable directions
3.1 Introduction
3.2 Models and background
3.2.1 Bootstrap percolation
3.2.2 Kinetically constrained models
3.2.3 Result
3.3 Sketch of the proof
3.4 Preliminaries and notation
3.5 Droplet algorithm
3.5.1 Clusters and crumbs
3.5.2 Distorted Young diagrams
3.5.3 Span
3.5.4 Droplet algorithm and spanned droplets
3.5.5 Properties of the algorithm
3.6 Renormalised East dynamics
3.6.1 Geometric setup
3.6.2 Arrow variables
3.6.3 Renormalised East dynamics
3.6.4 Proof of theorem 3.8
3.7 Open problems
4 Convergence to equilibrium in supercritical kinetically constrained models 
4.1 Introduction
4.2 Notations and result
4.3 Dual paths
4.4 Codings
4.5 An auxiliary process
4.5.1 Local spread of zeroes
4.5.2 Definition of the auxiliary process
4.5.3 Properties of the auxiliary process
4.6 Proof of proposition 4.11
5 Convergence to equilibrium in the d-dimensional East model
5.1 Introduction
5.2 Notations and results
5.3 Proof of theorem 5.2
5.3.1 Finding a site that stays at zero for a time (t)
5.3.2 Proving the origin stays at zero for a time (t)
5.3.3 Ending the proof of theorem 5.2
5.4 Proof of corollary 5.4

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