arXiv:1910.06782v1 [math.PR] 15 Oct 2019
IVAILO HARTARSKY, FABIO MARTINELLI, AND CRISTINA TONINELLI ABSTRACT. In this paper we consider kinetically constrained mod-els (KCM) on Z2with general update familiesU. For U belonging
to the so-called “critical class” our focus is on the divergence of the infection time of the origin for the equilibrium process as the density of the facilitating sites vanishes. In a recent paper [11] Marˆech´e and two of the present authors proved that ifU has an in-finite number of “stable directions”, then on a doubly logarithmic scale the above divergence is twice the one in the corresponding U-bootstrap percolation.
Here we prove instead that, contrary to previous conjectures [15], in the complementary case the two divergences are the same. In particular, we establish the full universality partition for critical U. The main novel contribution is the identification of the leading mechanism governing the motion of infected critical droplets. It consists of a peculiar hierarchical combination of mesoscopic East-like motions. Even if each path separately depends on the details of U, their combination gives rise to an essentially isotropic motion of the infected critical droplets. In particular, the only surviving information about the detailed structure ofU is its difficulty. On a technical level the above mechanism is implemented through a sequence of Poincar´e inequalities yielding the correct scaling of the infection time.
2010 Mathematics Subject Classification. Primary 60K35; Secondary 82C22, 60J27, 60C05.
Key words and phrases. Kinetically constrained models, bootstrap percolation, universality, Glauber dynamics, Poincar´e inequality.
Financial support: ERC Starting Grant 680275 “MALIG”, ANR-15-CE40-0020-01 and PRIN 20155PAWZB “Large Scale Random Structures”.
1. INTRODUCTION
1.1. Kinetically constrained models. We directly define the mod-els of interest and refer the reader to the companion paper [11] for more background. Let U be a finite collection of finite subsets of Z2 \ {0} called update rules and consider the following interacting particle systems onΩ = {0, 1}Z2
parametrised byU and q ∈ (0, 1). On each site x ∈ Z2 we are given an independent Poisson clock of pa-rameter one and at each arrival timet := tx,k of the clock the process attempts to update the current stateωx(t) according to the following rule. If the configurationω(t) is such that there exists U ∈ U such that ∀y ∈ U ωx+y(t) = 0, then ωx(t) is resampled from the Bernoulli(1 − q)-measureµq(1) = 1−q, µq(0) = q. In this case we say that a legal update occurs at timet at site x. Otherwise the attempted update is rejected. Using the fact that all rings {tx,k}x∈Z2,k∈N of the Poisson clocks are different a.s. and that all updating rules are finite sets, it is easy to check that the above process is well defined [13]. Moreover, since the update rules do not contain the origin, the process is reversible w.r.t. the product measureµ := µZ2
q .
We will refer to the above process as the kinetically constrained spin
model with update family U, for short U-KCM or just KCM if U is
clear from the context. KCM have been introduced several years ago in the physics literature (but only for certain specific choices of U) in order to reproduce in simple and fundamental interacting particle systems some of the main features of the so-called glassy dynamics,
i.e. the dynamics of a supercooled liquid near the glass transition
[2, 8, 9, 12, 18].
1.2. Bootstrap percolation. TheU-KCM can also be seen as the non-monotone stochastic counterpart of theU-bootstrap percolation mono-tone cellular automaton onΩ (see e.g. [4, 17]). In the latter process one says that x ∈ Z2 is infected for ω ∈ Ω if ω
x = 0 and healthy oth-erwise and the relevant time evolution concerns the set of infected sites At at integers times t. Given At, the set At+1 is constructed by adding toAtany healthy sitex for which there exists U ∈ U such that U + x ⊂ At:
At+1 = At∪ {x ∈ Z2, ∃ U ∈ U, x + U ⊂ At}.
The only randomness in the process occurs at time t = 0 by taking the initial infection as the random setAω = {x ∈ Z2 : ωx = 0} with ω ∼ µ. If At=0 = A one writes [A]U = St>0At for the closure of A under theU-bootstrap process.
(e.g. in mean, median, or w.h.p.) of the infection time of the origin defined as
τ0 = inf{t : ω0(t) = 0}.
Notice that for bootstrap percolationτ0 ∈ N ∪ {+∞} and it only de-pends also on the initial set of infection Aω. On the other hand for the KCMτ0 ∈ [0, +∞] and it depends on the occurrences of the Pois-son processes at the vertices of Z2 and on the coin tosses used for the various legal updates. In order to present our main result on τ0 we need some further notation (see [5]).
For each unit vectoru ∈ S1, let H
u := {x ∈ R2 : hx, ui < 0} denote the half-plane whose boundary is perpendicular tou.
Definition 1.1 (Stable directions). The set of stable directions ofU is S = S(U) = u ∈ S1 : [Hu∩ Z2]U = Hu∩ Z2
.
Using the above definition the update familyU was classified in [5] as:
• supercritical if there exists an open semicircle in S1 that is disjoint from S,
• critical if there exists a semicircle in S1that has finite intersec-tion withS, and if every open semicircle in S1 has non-empty intersection withS,
• subcritical if every semicircle in S1 has infinite intersection withS.
In [5] together with [1] it was also proved that for all q ∈ (0, 1) τ0 < +∞ a.s. for the U-bootstrap percolation process iff U is either supercritical or critical. The next definition quantifies the difficulty of propagation of infection in a stable direction for bootstrap perco-lation.
Definition 1.2 (Definition 1.2 of [4]). LetU be an update family and u ∈ S1 be a direction. The difficulty ofu, α(u), is defined as follows.
(i) Ifu is unstable, then α(u) = 0.
(ii) Ifu is an isolated (in the topological sense) stable direction, then α(u) = min{n ∈ N : ∃K ⊂ Z2, |K| = n, |[Z2 ∩ (Hu∪ K)] \ Hu| = ∞},
i.e. the minimal number of infections allowing Hu to grow in-finitely.
The difficulty ofU is
α(U) = inf
C∈Csupu∈Cα(u), (1)
whereC is the set of open semi-circles of S1.
Remark 1.3. It was proved in [5, Lemma 5.2] (see also [4, Lemma 2.7]) that1 6 α(u) < ∞ if and only if u is an isolated stable direction. A key result of [4] states that ifTU denotes the median ofτ0 for the U-bootstrap process then for any critical U with difficulty α
lim q→0
log log(TU)
log(1/q) = α. (2)
1.3. Main results. Our main result is that (2) holds also for the U-KCM if S is finite. The core of the proof is based on the discovery of a new and efficient relaxation mechanism completely different from the one occurring in bootstrap percolation. While for the latter the dominant mechanism to grow infection is a linear expansion from some rare large groups of infected sites (critical droplets), for KCM we find that these droplets, in order to move around in an efficient way to infect the origin, perform a complex hierarchical motion (see Section 2 for an heuristic detailed description). The above motion is a novel type of relaxation mechanism with respect to all those con-sidered so far in the KCM literature. In particular, it is different from the random walk like motion that captures the dominant behavior for 2-neighbour model [16], and it is also different from the purely East-like motion used to establish the scaling for models with an in-finite number of stable directions [11, 14, 15]. Indeed, based on the wrong intuition that the two former mechanisms were essentially the only two possible efficient ways to move critical droplets around, a conjecture was put forward in [15, Conjecture 3], which is disproved by our result, Theorem 1 below.
Write Eµ(τ0) for the expectation of the infection time for the U-KCM with initial lawµ (i.e. for the stationary process).
Theorem 1. Let U be a critical update family with finite stable set S
and difficultyα. Then
Eµ(τ0) = eO(log(1/q)3/qα). (3) Moreover, lim q→0 log log(Eµ(τ0)) log(1/q) = α. (4)
with [16, Lemma 4.3] and (2). Theorem 1 together with [11, The-orem 2.8] and [15, TheThe-orem 2(a)] corrects [15, Conjecture 3] and gives the following full universality picture forU-KCM with critical U. Theorem 2. LetU be a critical update family. Then
lim q→0 log log(Eµ(τ0)) log(1/q) = ( α if|S| < +∞, 2α otherwise. (5)
Some key partial results in the direction of proving Theorem 2 were established in [15, 16]. In particular, in [15] the scaling (4) was proved for any update family U with maxu∈S1α(u) = α while [14] proved (5) with a higher degree of precision for a specific model with |S| = ∞, the Duarte model. We refer the interested reader to the in-troduction of [11] for a detailed account of the history leading to (5). For supercritical update families the correct scaling was determined in [15, Theorem 1] and [14, Corollary 4.3].
1.4. Organisation of the paper. We start by providing in Section 2 a heuristic explanation of the relaxation mechanism underlying our main result. In section 3 we fix some notation and gather some pre-liminary tools from bootstrap percolation that are by now well es-tablished in the literature. We will not dwell on the technical as-pects of the definitions and invite the reader to refer to section 4.3 of [15], which we follow closely, for more details. For reader’s conve-nience we have collected in section 3.2 three useful technical lemmas on certain one-dimensional kinetically constrained Markov processes. Although the proof of these lemma can be found or derived from the existing literature on KCM, we have added it in the appendix for completeness. Section 4 contains the main new technical Poincar´e inequality, while Theorem 1 is proved in Section 5. Finally, in Section 6 we discuss some natural open problems raised by the present work.
2. SOME HEURISTICS BEHIND THEOREM 1
For a high-level and accessible introduction to the main general ideas and techniques involved in bounding from above Eµ(τ0) we re-fer to [15, Section 2.4]. There, in particular, it was stressed that while the necessary intuition is developed using dynamical considerations (e.g. by guessing some efficient mechanism to create/heal infection inside the system), the actual mathematical tools are mostly analytic and based on suitable (and, unfortunately, sometimes very technical) Poincar´e inequalities. This paper makes no exception.
0
0 0
0
(A) The four update rules
belong-ing toU.
1 2
2
2
(B) The difficulties of the four
sta-ble directions of the model.
FIGURE 1. An example of an update family U with
fi-nite number of stable directions examined in Section 2.
In order to go beyond the results of [15] and get the sharp scal-ing of Theorem 1 in the case of a finite set of stable directions, the following new key input is needed.
For simplicity imagine that U has only four stable directions coin-ciding with the four natural directions of Z2. For a generic model with |S| < ∞ the mechanism is the same, the only difference being that in general ‘droplets’ have a more complex geometry. Assume further thatα(~e1) = 1 and α(−~e1) = α(±~e2) = 2 (see Figure 1). Consider now a critical droplet, i.e. a square frameD, centered at the origin, of side length≈ C log(1/q)/q, C ≫ 1, and O(1)-thickness, and suppose that D is infected. Then, w.h.p. (w.r.t. µ) there will be extra infected sites next to D in the ~e1-direction allowing D to infect D + ~e1. However, it will be extremely unlikely to find a pair of infected sites near each other and next to the other three sides of D because of the choice of the side length of D. We conclude that w.h.p. it is easy for D to advance forward in the ~e1-direction but not in the other directions. Moreover, as explained in detail in [15, Section 2.4], an efficient way to effectively realize the motion in the~e1-direction is via a generalised East path. In its essence the latter can be described by the following game. At every integer time a token is added or removed (if already present) at some integer point according to the following rules:
(i) each integer can accomodate at most one token; (ii) a token can be freely added or removed at1;
(iii) for anyj > 2 the operation of adding/removing a token at j is allowed iff there is already a token atj − 1.
Givenn ∈ N, by an efficient path reaching distance n we mean a way of adding tokens to the original empty configuration to finally place
C log(1/q)/q2 C
q log( 1 q)
(A) The infected droplet (black frame with width O(1)) progressively moves
to the right in an East-like way using the extra infected sites present w.h.p. in each column of the gray rectangle. This progression stops when reaching the first infected horizontal pair of sites at the correct height (red pair).
(B) The infected droplet on the right grows into an e2-extended droplet
thanks to the infected pair of sites. The movement is then reverted, pro-gressively retracting the extended droplet in an East way until reaching the original position.
FIGURE 2. The mechanism for the droplet to grow in the~e2-direction.
one at n which uses a minimal number of tokens. A combinatorial result (see [7]) says that the optimal number grows likelog2(n).
The main new idea now is that, while w.h.p. the droplet D will not find a pair of infected sites (which are necessary to grow an extra layer of infection in the ~e2-direction) next to e.g. its top side, w.h.p. it will find it at the right height within distance C log(1/q)/q2 in the ~e1-direction (see Figure 2). Hence, a possible efficient way for D to move one step in the~e2-direction is to:
(A) travel in the~e1-direction in a East-like way until finding the nec-essary pair of infected sites within distance C log(1/q)/q2 from the origin;
(B) grow there an extra layer in the ~e2-direction and retrace back to its original position while keeping the acquired extra layer of infection.
A similar mechanism applies to the −~e2-direction. Slightly more in-volved is the way in which D can advance in the −~e1-direction. In this case the extra infected pair needs to be found within distance C log(1/q)/q2 from the origin in the vertical direction (see Figure 3). In order to reach it,D performs an East-like movement upwards, each
C
q2 log(1/q)
FIGURE 3. The mechanism for the droplet growth in
the −~e1-direction. The droplet moves in an East way in the~e2-direction by making long excursions in the~e1 -direction as in Figure 2.
of whose steps is itself realised by the back-and-forth East motion in thee1 direction described above.
Using the result for the typical time scales of the generalised East process (see [15, formula (3.5)]) it is easy to see that the typical excursion of D for a distance ℓ ≡ C log(1/q)/q2 in the ~e
1-direction requires a time
Tloc = q−|D|O(log(ℓ)) = eO(log
3(1/q))/q .
This time scale also bounds from above the time scale necessary to advance by one step in the “hard” directions−~e1, ±~e2.
In conclusion, by making a “quasi-local” (i.e on a length scale ℓ) East-like motion in the easy direction~e1, the infected critical droplet D can actually perform a sort of random walk in which each step requires a timeTloc. The result of Theorem 1 becomes now plausible provided that one proves that anomalous regions of missing helping infected sites do not really constitute a serious obstacle.
The above dynamic heuristics can be turned into a rigorous argu-ment using canonical paths. However, a much neater approach is to prove a Poincar´e inequality for the U-KCM restricted to a suitable
finite domain of Z2 (see Theorem 4.5). More precisely, in the toy ex-ample discussed above the inequality that we establish is as follows.
Let V = T ∪ T0 ∪ T1− ∪ T1+ where T , T0, T1± are as in Figure 4. The ratio of the sides of the rectangle T is Θ(q) while for the other rectangles it isΘ(1).
Clog(1/q)q2
Clog(1/q)q T
T0
T1− T1+
D
FIGURE 4. The geometric setting (denoted “snail” in Section 4) for the toy model of Figure 2
LetΩ0 consist of all configurations of{0, 1}V such that: (i) each column ofT contains an infected site;
(ii) each row ofT0 contains a pair of adjacent infected sites; (iii) each column ofT1± contains a pair of adjacent infected sites. Notice that by choosing C large enough µ(Ω0) = 1 − o(1) as q → 0. Then, in the key Theorem 4.5, we prove that for anyf : {0, 1}V → R
1{D is infected}VarV(f | Ω0) 6 e
O(log(1/q)3)/q
× D(f ),
where D(f ) is the Dirichlet form of f (see (8)). One can interpret the above inequality as saying that the KCM in V restricted to the good setΩ0 has a relaxation time at mosteO(log(1/q)
3)/q
. We prove this by an inductive procedure over T0, T1± which, in some sense, makes rigorous the dynamic heuristics described above.
3. NOTATION AND PRELIMINARIES
In this section we gather the relevant notation and basic inputs from bootstrap percolation and KCM theories.
3.1. Bootstrap percolation.
3.1.1. Stable and quasi-stable directions. We often refer to vertices of the lattice Z2 as sites. For every integern, denote [n] := {0, 1, . . . , n − 1}. We fix a critical update family U with difficulty α = α(U) and with a finite setS of stable directions1. Using the definition of α(U) (see (1)) one can fix an open semicircle C with midpoint u0, one of 1By Lemma 2.6 of [4] or Theorem 1.10 of [5] this is equivalent to the fact that
whose endpoints is in S and such that maxu∈Cα(u) = α. Using [5, Lemma 5.3] (see also [4, Lemma 3.5] and [15, Lemma 4.6]) one can choose a set of rational directions S′ ⊃ S, so that for every two consecutive elementsu and v of S′ there exists an update ruleX ∈ U such thatX ⊂ (Hu∪ ℓu) ∩ (Hv∪ ℓv), where ℓu′ = {x ∈ R2 : hx, u′i = 0} for any u′ ∈ S1. The elements of S′ are usually referred to as
quasi-stable directions. Then our fundamental set of directions will be
ˆ
S = [
u∈S′
({u, u0− (u − u0)} + {0, π/2, π, 3π/2}). (6) In other words, we start with the stable directions, add to them the quasi-stable ones, reflect them atu0 and finally make the set obtained invariant by rotation byπ/2. By construction the cardinality of ˆS is a multiple of 4.
Remark 3.1. Let us note that invariance by rotation and reflection is cosmetic and one could in fact deal directly with the set of quasi-stable directions from [5], though notation would be more laborious and drawings less aesthetic.
We write u0, u1, . . . u4k−1 for the elements of ˆS ordered clockwise starting with u0. For lightness of notation we will also denote by ui the unit vector in R2 pointing in the ui-direction. For all figures we shall take ˆS = {iπ/4, i ∈ {0, 1, . . . , 7}} and u0 = π. When referring to ui, the indexi will be considered modulo 4k. In particular, with this convention, the directions u−k+1, . . . , uk−1 will be the only elements of ˆS belonging to the open semicircle C.
Given s > 0 and ui, uk ∈ ˆS we set Hui(suk) := Hui + suk and ℓui(suk) = ℓui+ suk. We write Hui(suk) := Hui(suk) ∪ ℓui(suk) for the closure of Hui(suk). If uk = uithen we shall simply write Hui(s), ℓui(s) etc. We write ρi = inf{ρ > 0, ∃x ∈ Z2, hx, uii = ρ} for the smallest positive s such that ℓui(s) 6= ℓui and ℓui(s) ∩ Z
2 6= ∅. Since all the quasi-stable directions are rationalρi > 0 for all i.
3.1.2. Geometric setup. We next turn to defining the various geomet-ric domains we will need to consider. As the notation is a bit cum-bersome, the reader is invited to systematically consult the relevant figures. We fix a large integer w and a small positive number δ de-pending onU (e.g. w much larger than the diameter of U and of the largest difficulty of stable directions), but not depending onq. When using asymptotic notation (asq → 0) we will assume that the implicit constants do not depend on W, δ and q. Throughout the entire paper we shall consider thatq is small, as we are interested in the q ↓ 0 limit.
R3− w u0
w
FIGURE 5. The shaded region is the quasi-stable ring
R, while the hatched one is the quasi-stable half-ring H. As anticipated all the radii Ri are much larger than the widthw.
In particular, we shall assume thatq is so small that any length scale diverging to+∞ as q ↓ 0 will be (much) larger than the constant w. Definition 3.2. Consider a closed convex polygon in R2 with m 6 4k sides. Assume that the outward normal vectors to the sides of P belong to ˆS and that ui is one of them. Then we write ∂iP (or sometimes∂uiP ) for the side whose outward normal is ui.
We can now define the notion of droplet that will be relevant for our setting (see Figure 5). Recall the definition of the open semicircle C.
Definition 3.3 (The quasi-stable ring and half-ring). Fix a radiusR = R(q) such that limq→0R(q) = +∞ and let Ri = ρi
j R ρi k
, i ∈ [4k]. We call the convex set of R2
R := \ i∈[4k] Hu i(Ri) \ \ i∈[4k] Hu i(Ri− w)
the quasi-stable ring (or simply the ring) with radiusR and width w centered at the origin. We writeRint for the regionT
i∈[4k]Hui(Ri− w) enclosed by R. Clearly the outer boundary of R is a closed convex polygon P satisfying the assumption of Definition 3.2 and we write ∂iR for ∂iP . We also let
∂CR := k−1[ i=−k+1 ∂iR, and we call H = k \ i=−k Hu i(Ri) \ k−1\ i=−k+1 Hu i(Ri− w)
H + Lu0 R T± 0 T T1+ T− 1 T− 2 T + 2 r0 r1 r1 r2 r2 L
FIGURE 6. A snail with its tube T and its trapezoids
T±
1 , T2±, . . . . The right-snail V+consists of T and of T+
i , i > 1, while the left-snail of T and of Ti−, i > 1. In Theorem 4.5 the shaded quasi-stable ring R and half-ringH + Lu0 will be assumed to act as an infected boundary condition.
the quasi-stable half-ring of radiusR and width w.
Our approach will consist in building progressively larger domains for which we can bound the Poincar´e constant of the finite volume KCM process conditionally on the simultaneous occurrence of a cer-tain likely event and the presence of an infected ring. We next define these domains (see Figure 6). Recall that δ is a small constant de-pending on the update familyU.
Definition 3.4 (Snails). LetL = L(q) > 0 be such that limq→0L(q) = +∞ and assume that L(q)hu0, uk−1i/ρk−1 ∈ N. We call a sequence of non-negative numbers r= (r0, r1, . . . , r2k) admissible if
0 6 r0 6δL, ri 6δri−1, r2k = 0. Given an admissible r we call the set
VL+(r) = k−1\ i=−k+1 Hu i(Ri+ Lu0) ∩ 3k \ i=k Hu i(Ri+ ri−k)
the right-snail with parameters (L, r). Using the symmetric construc-tion of ˆS, the left-snail VL−(r) with parameters (L, r) is simply defined as the reflection of the right-snail w.r.t. the line orthogonal tou0 and passing through the point 1
2Lu0. Finally the snail with parameters (L, r) is the set
VL(r) = VL+(r) ∪ VL−(r).
We systematically drop the parameter r from our notation when no ambiguity arises.
of as the set obtained by stacking together as in Figure 6 the tube T = VL((0, . . . , 0))
and the trapezoids.
Ti+ = VL+(r0, . . . , ri, 0, . . . , 0) \ VL+(r0, . . . , ri−1, 0, . . . , 0) = (Huk+i(Rk+i+ ri) \ Huk+i(Rk+i))
∩ Huk+i−1(Rk+i−1+ ri−1) ∩ Huk+i+1(Rk+i+1) With this picture in mind the positive values of r coincide with the heights of the corresponding non-empty trapezoids. A similar decomposition holds for the left-snail.
Definition 3.6. Fix i ∈ [2k] and r such that ri > 0. Then we set L+j,i = Ti+ ∩ ℓui(Ri + jρi) ∩ Z
2 in such a way that the ith-trapezoid can be decomposed as T+ i ∩ Z2 = S j>0L+j,i ∩ Z2. Similarly, if ◦ T := T \ (R ∪ Rint∪ (H + Lu 0)), we can decompose ◦ T asT ∩ Z◦ 2 =S j>0Cj, whereC′ 0 = ∂CR, Cj+1′ = ◦ T ∩ (C′
j+ inf{ρ > 0, (Cj′ + ρu0) ∩ Z2 6= 0}u0), andCj = Cj′ ∩ Z2 forj > 0.
Note that for sufficiently smallδ and any admissible sequence r the sets {L+
j,i}j>0 are either empty or are disjoint segments containing at least δ2kL sites. Similarly, in each segment belonging to C
j the number of lattice sites is either zero or Ω(R). In the sequel we will only consider non-emptyL+
j,i andCj without explicitly specifying the range ofj > 0.
3.1.3. Helping sets. If u is a stable direction, then an infected rectan-gle oriented orthogonally to u needs some ‘help’ from other infected sites in order to propagate its infection in the u-direction. Our next ingredient provides us with a sufficient condition for this to happen. It is the content of [4, Lemmas 3.2 and 3.4] and [5, Lemma 5.4]. For notational convenience, given A ⊂ R2 we shall write[A]
U for the U-closure of A ∩ Z2.
Lemma 3.7. Fixu = ui, i ∈ [4k], with difficulty α(ui).
(a) LetZ ⊂ Z2\Hube a set of cardinalityα(u) such that [Hu∪Z]U∩ℓu
is infinite. There exist a1, . . . , am, b ∈ ℓu such that the following holds
for all w large enough and r > w2 large enough. Let
ui−1
ui ∂iT
ui+1
FIGURE 7. The geometric setting of Lemma 3.7. The
hatched trapezoid is T and the consecutive infected sites can be a possible helping set.
be the trapezoid in Figure 7 and letk1, . . . , km ∈ Z be such that for every j ∈ [m] the site aj+ kjb belongs to ∂iT and is at distance at least w from
the endpoints of∂iT . Let A =Smj=1(Z + aj+ kjb). It holds that if T and A are infected, then the U-bootstrap map restricted to
Hi−1(r) ∩ Hi(w/2) ∩ Hi+1(r) ∩ Hi+2k(w)
infects ∂iT .
(b) The same conclusion holds withA replaced by w consecutive sites
in∂iT and the U-bootstrap map restricted to T ∪ ∂iT .
Definition 3.8 (u-helping sets). Let i ∈ [−k + 1, k − 1]. Then we call an infected setA as in part (a) of the above lemma a ui-helping set. 3.2. Some KCM tools. For reader’s convenience we next collect some general tools from KCM theory that will be applied several times throughout the proof of the main result.
3.2.1. Notation. For every statement P define 1{P} = 1 if P holds
and 1{P} = 0 otherwise. For any subset Λ of R
2 we write (Ω Λ, µΛ) for the product probability space ({0, 1}Λ∩Z2
, ⊗x∈Λ∩Z2µq). If Λ = R2, we simply write (Ω, µ). Given f : ΩΛ → R we shall write µΛ(f ) and VarΛ(f ) for the mean and variance of f w.r.t. µΛ respectively whenever they exist. For any elementω ∈ Ω and Λ ⊂ R2 we writeω
Λ for the collection{ωx}x∈Λ∩Z2. Given a functionf : Ω → R depending on finitely many variables we write
D(f ) = X x∈Z2
µ(cxVarx(f )), (8)
for the KCM Dirichlet form of f , where cx(ω) is the indicator of the event {∃U ∈ U : ωx+y = 0 ∀y ∈ U} and Varx(f ) denotes the condi-tional variance Var(f | {ωz}z6=x). Finally, we shall write Pµ(·) for the
expectation w.r.t. Pµ(·).
3.2.2. Poincar´e inequalities. We begin with a well-known general fact on product measures which we state here in ready-to-use form. Lemma 3.9. Let Λi, i ∈ {1, 2, 3} be three disjoint finite subsets of Z2
andνi be a probability measures on ΩΛi. Let ν be the product measure
onΩΛ, whereΛ =SiΛi. Then for any local functionf we have ν1(VarΛ2∪Λ3(f )) 6 VarΛ(f ) 6 ν1(VarΛ2∪Λ3(f )) + ν2(VarΛ1∪Λ3(f )).
Proof of Lemma 3.9. The total variance formula reads
VarΛ(f ) = ν1(VarΛ2∪Λ3(f )) + VarΛ1(νΛ2∪Λ3(f )).
The first inequality then follows immediately. For the second one we observe that VarΛ1(νΛ2∪Λ3(f )) = ν1((νΛ2∪Λ3(f − ν(f )) 2) 6ν((f − νΛ1∪Λ3(f )) 2) = ν 2(VarΛ1∪Λ3(f )) by Jensen’s inequality.
The other results take the form of a constrained Poincar´e inequality for an appropriate probability space and can be interpreted as pro-viding an upper bound on the relaxation time of a suitable reversible Markov process with certain kinetic constraints.
Lemma 3.10. Let (Ωi, νi), i = 1, 2 be two finite probability spaces and
let(Ω, ν) be the associated product space. Let also E ⊂ Ω1 withν1(E) > 0. Then
Varν(f ) 6 2ν1(E)−1ν Varν1(f ) +1{ω
1∈E}Varν2(f )
.
Proof of Lemma 3.10. It follows from [6, Proof of Proposition 4.4]
that Varν(f ) 6 1 1 −p1 − ν1(E) ν Varν1(f ) +1{ω 1∈E}Varν2(f ) 62ν1(E)−1ν Varν1(f ) +1{ω 1∈E}Varν2(f ) .
The final result concerns certain generalisations of the basic 1-neighbour KCM process, commonly known as FA1f KCM (see also [15, Section 3.1]).
Let( ˆS, ˆν) be a finite probability space with ˆν a positive probability measure and let Ωn = ˆS[n] and ν = Ni∈[n]νi, where νi = ˆν for all i ∈ [n]. Given A ⊂ ˆS and an integer k < n, let
where κA,±i is the indicator of the event thatωj ∈ A for all j ∈ {i ± 1, . . . , i ± k}. We either view indices modulo n (we view [n] as the n-cycle) or we make the convention that κA,±i = 0 if i ± k /∈ [n]. Remark 3.11. In applications the integer k will be a large constant not depending onq, ˆS = {0, 1}m form ≪ n potentially depending on q, and ˆν will be the Bernoulli(1 − q) product measure conditioned on some very likely event.
Lemma 3.12. Assume that (1 − ˆν(A)k)n/(3k) < 1
16 and denote ΩA = S
i∈[n]{ω | κAi (ω) = 1}. Then, for all f : Ωn → R Varν(f | ΩA) 6 (2/ˆν(A))O(k)
n X i=1 ν κAi Varνi(f ) . (9)
The proof is left to the appendix.
4. THE CORE OF THE PROOF
In this section we prove a Poincar´e inequality which will represent the key step in the proof of Theorem 1. In order not to obscure the main ideas and techniques with unnecessary technical complications (mostly related to the geometry of the quasi-stable ring) we shall do so under the following simplifying assumption (see also [15, Assump-tion 6.1].
Assumption 4.1. For every stable direction u ∈ C there exists a sub-set Zu of the line ℓu of cardinality α such that [Hu ∪ Zu]U ∩ ℓu has infinite cardinality.
The proof applies directly without this assumption up to changing Gj in Definition 4.2 following [15, Sec. 7]. We will spare the reader the tedious details, as they already appeared previously in the above mentioned paper.
Given a snail V we shall work in the associated probability space ΩV = {0, 1}V ∩Z
2
endowed with the probability measureµV(· | G) con-ditioned to the simultaneous occurrence of the following events on ΩV.
Definition 4.2 (Good events).
(a) For each Cj let Gj be the event that for all directions ui ∈ C the part ofCj orthogonal toui contains an infectedui-helping set unless it is empty.
(b) For each non-empty L+j,i let G+j,i be the event that there exist w consecutive infected sites inL+j,i.
Using the above events we then define G◦ T = \ j Gj, GT = GR∩ G◦ T, G + i = \ j Gj,i. Finally we set GV+ = GT ∩T i∈[2k]G + i and G := GV = GV+ ∩ GV−, withGV− the analog ofGV+ for the left snail.
In the sequel we will mostly abbreviate the notation by setting G±:= G
V±.
Remark 4.3. The events above are defined so as to preserve as much as possible the original product structure ofµ in the conditional mea-sureµV(· | G). In fact, µV±(· | G±) = µT(· | GT) ⊗ O i∈[2k] µT± i (· | G ± i ) , µT± i (· | G ± i ) = O j µL+ j,i(· | G ± j,i), µT(· | GT) = δωR∪(H+Lu0)=0⊗ µRint⊗ O j µCj(· | Gj) .
Further note that if the length of the smaller base of thei-th trapezoid T±
i is at leastΩ(q−2w) then µ(G±i ) > 1 − o(1). In particular, that holds if r is admissible and ri−1 > q−2w. Similarly, if R > w2log(1/q)/qα andL 6 q4w, we have µ(G◦
T) > 1 − o(1) (see e.g. [15, Lemma 6.5]). Finally,µ(GR) = q|R∪(H+Lu0)| = qΘ(Rw), so that if R > w2log(1/q)/qα, L 6 q4w, r is admissible and for some i we have r
i+1 = 0 and ri > q−2w, then
µ(G) = qΘ(Rw). (10)
Finally, we observe that the requirement thatH + Lu0 is infected in the definition of the eventGRis there only to ensure the easy removal of the simplifying Assumption 4.1.
Before stating our main result we make the following simple obser-vation.
Lemma 4.4 (Ergodicity lemma). For anyω ∈ G+
{x ∈ V+ : ωx = 0} V+
U = V
+∩ Z2.
Proof of Lemma 4.4. Let ω ∈ G+ ⊂ Ω
V+. It is easy to see that R infects Rint, since there exist unstable directions, asU is critical. We then proceed by induction onL to show that T becomes infected by
decomposing it into Cj, using Lemma 3.7 and the fact that ω ∈ GT. Thereafter, by induction on r (on the last non-zero coordinate and on the number of non-zero coordinates), decomposing Ti+ into L+j,i, using Lemma 3.7 and the fact that ω ∈ G+j,i, we conclude that V+
becomes infected as desired.
After this warm-up we are ready to state the main result of this section. In the sequel, for anyΛ ⊂ Z2, any x ∈ Λ and any ω
Λ ∈ ΩΛwe shall write cΛ
x(ωΛ) for the constraint cx(ω) computed for the configu-rationω equal to ωΛ in Λ and equal to 1 elsewhere. By construction, cΛ
x(ωΛ) 6 cx(ω′) for any ω′ ∈ Ω such that ωΛ′ = ωΛ. Then for any snail V (or tube) we write γV for the best (possibly infinite) constantγ > 1 such that the Poincar´e inequality
VarV(f | GV) 6 γ X x∈V µV cVx Varx(f ) (11) holds.
Theorem 4.5. There existw0, δ0 > 0 such that for any 0 < δ 6 δ0 and w > w0 the following holds for any R > w2log(1/q)/qα. Consider the
snail VL(r) for admissible L, r such that r2k−1 > q−2w and L 6 q−4w.
Then
γV 6q−O(Rw log L). (12)
Notice that (12) follows from the analogous statement withV and G replaced by (V±, G±). Indeed, one can apply Lemma 3.9 to V+\ (T ∪ T0), V−\ (T ∪ T0) and T ∪ T0.
Hence, in the sequel we will concentrate on proving (12) for the best constantγV+ in the Poincar´e inequality (11) withV replaced by its right-snailV+and the good eventG replaced by the good event G+ for V+. The proof is based on comparison methods between Markov processes and induction over right-snails with different L and r fol-lowing the one in Lemma 4.4. If we exchange right-snails with left-snails the same proof will then apply to the left-snailV−as well. Since our arguments never require a left-snail, for lightness of notation, we drop the superscript “+” from our notation whenever possible.
The proof of the theorem is decomposed into two quite different steps (see Propositions 4.7 and 4.9 below). A first one, labeled the
base case, in which we consider a right snail V with no trapezoids (r =0). In the second step, labeled reduction step, roughly speaking we compare the Poincar´e constantγV of a generic right snailV with the same constant computed for V stripped off its trapezoids.
The conclusion of Theorem 4.5 follows at once from (10), Proposi-tion 4.7 and ProposiProposi-tion 4.9.
is very important to emphasise that it is only in the first step that we use the very definition of the event GT entering in the event G (cf. Definition 4.2). In the second step the only property of the eventGT that is needed is that it is a decreasing event in ΩT w.r.t. the partial orderω ≺ ω′ iffω
x 6ωx′ ∀x ∈ T .
In the sequel fix δ, w, R as in the statement of the theorem and recall thatT = VL(0).
The base case.
Proposition 4.7. For anyf : ΩT → R VarT(f | GT) 6 q−O(Rw log L)
X x∈T
µT(cTx Varx(f )).
Proof of Proposition 4.7. We first observe that, up to minor
modifica-tions, in [15, Proposition 6.6] it was proved that for allf : Ω◦
T → R 1G RVar◦ T(f | G ◦ T)) 6 q −O(Rw log L) 1G R X x∈ ◦ T µ◦ T(c T x Varx(f )). (13) The next step in the proof is an analogous result forRint.
Lemma 4.8. For anyf : ΩRint → R
1G RVarRint(f ) 6 q −O(Rw) 1G R X x∈Rint µRint(cxVarx(f )).
Proof of Lemma 4.8. Let us partition Rint into disjoint strips of width w parallel to the direction u0 which we order from bottom to top. We observe that ifR is infected then part (b) of Lemma 3.7 ensures that one can infect all the strips starting e.g. from the bottom one.
We can then apply [15, Proposition 3.4] on the generalised FA1f KCM to obtain
VarRint(f ) 6 q−O(Rw) X strip
µ((κ++ κ−) Var
strip(f )),
where κ± are the indicators of the events that the previous (resp. next) strip is fully infected and we use the convention thatκ±≡ 1 for the last/first strip. W.l.o.g. it then suffices to bound the generic term µ(κ+Var
strip(f )). But this can be done using Lemma 5.2 of [15] and the previous observation on the spreading of the infection fromR to
Using Lemma 3.9 with Λ1 = ◦ T , Λ2 = Rint, Λ3 = ∅ and ν1 = µ◦ T(· | G ◦
T), ν2 = µRint together with Lemma 4.8 and (13) we obtain
1G R(ω) Var◦ T ∪Rint(f | GT◦) 61G R µRint Var◦ T(f | G ◦ T) + µ◦ T(VarRint(f ) | G ◦ T) 6q−O(Rw log L)1G R X x∈ ◦ T µ◦ T ∪Rint(c T x Varx(f )) + X x∈Rint µ◦ T(cxVarx(f ) | G ◦ T) . Usingcx = cTx for allx ∈ Rinttogether with Remark 4.3 we get imme-diately get that
VarT(f | GT) = µT(Var◦ T ∪Rint(f | GT◦) | GR) 6µ(GR)−1q−O(Rw log L) X x∈T µT(cTx Varx(f ))
and the proposition follows.
The reduction step.
Proposition 4.9. Consider the right snail V := VL(r) for admissible L, r such that r2k−1>q−2w. Then
γV 6 ew 3 / min ˆ V µ(GVˆ) O(log L) max ˆ T γTˆ,
whereminVˆ runs over all right snails VLˆ(ˆr) such that for some i ∈ [2k]
we have: (a)ˆri+1 = 0, (b) for all j 6 i it holds that 0 6 rj− ˆrj = O(ri),
and (c)0 6 L − ˆL = O(ri). Similarly maxTˆ runs over snailsVLˆ(0) (i.e.
tubes) with0 6 L − ˆL = O(r0).
Proof of Proposition 4.9. The first step of reduction consists in
remov-ing a trapezoid consistremov-ing of a sremov-ingle line. Recall the definition of ρi in Section 3.1.1.
Lemma 4.10 (Removing a single line). Let q−2w 6 L and r be an
admissible sequence such that for some i ∈ [2k] we have that ri−1 > q−2w, r
i = λρk+i for λ ∈ N and ri+1 = 0. In other words the last
trapezoid Ti consists of λ segments orthogonal to ui+k. Then, setting er = (r0, . . . , ri−1, 0, . . . , 0), eV = VL(er) and V = VL(r), we have
γV 6 Lq−w
4O(λ) γVe.
forλ = 1, in which case the last trapezoid of the right snail is Ti = L1,i. UsingGV = GVe ∩ G1,i, for anyf : ΩV → R Lemma 3.9 gives
VarV(f | GV) 6 µV VarVe(f | GVe) + VarL1,i(f | G1,i) | GV
. (14)
The first term in the r.h.s. above satisfies µV(VarVe(f | GVe) | GV) 6 γVe µL1,i(G1,i) µV X x∈ eV cVx Varx(f )
by the definition (11) of γVe and the fact that c e V
x 6 cVx. To bound the second term we will use Lemma 3.12 applied toµL1,i(· | G1,i) with k = w and constraining event A = {0} ⊂ {0, 1}. We get
VarL1,i(f | G1,i) 6 q
−O(w) X x∈L1,i
µL1,i(1{G
x}Varx(f )), (15) whereGxis the event thatw consecutive sites immediately to the left or to the right of x in L1,i are infected. Plugging this back in (14), we see that we need to bound from above a generic term of the form µV 1{G
x}Varx(f ) | GV
, x ∈ L1,i. Using again Lemma 3.9 the latter is bounded from above by
µL1,i\{x} 1{G
x}VarV ∪{x}e (f | GVe)
/µ(Gx).
On the eventGx we can apply Lemma 3.10 to the product space (ΩVe, µVe(· | GVe)) ⊗ (Ω{x}, µx)
with constraining eventEx ⊂ ΩVe the existence of a infected translate Tx ⊂ eV of the trapezoid T = Hui+k−1(w 2) ∩ H i+k∩ Hui+k+1(w 2) ∩ H ui+3k(w)
directed by ui+k of height w and width larger than w, but not de-pending on q, such that x and the next w sites to the right and to the left of x in L1,i (those of them which exist) belong to ∂i+kTx (see Figure 7). Using |Tx ∩ Z2| = Θ(w4) and noticing that by the Harris inequality [10]µVe(Ex| GVe) > µVe(Ex), we get 1{G x}VarV ∪{x}e (f | GVe) 6 q −w4 µV ∪{x}e VarVe(f | GVe) +1{G x∩Ex}Varx(f ) | GVe . Finally, since ui+k is an isolated (quasi-)stable direction, it is easily seen (see Figure 7 and Lemma 3.7) that1{G
x∩Ex} 6c
V
definition of the Poincar´e constantγV given before Theorem 4.5, we conclude that µV(1{G x}VarV ∪{x}e (f | GVe) | GV) 6 q −O(w4) γVe X y∈ eV ∪{x} µV(cVy Vary(f )). Putting all together we finally get
VarV(f | GV) 6 |L1,i|q−O(w
4) γVeX x∈V µV cVx Varx(f ) ,
where the factor |L1,i| = O(L) comes from the fact that each vertex x ∈ L1,i produces a term of the form Py∈V µV(cVy Vary(f )). Recalling the definition (11) of γV the proof is finished. The remaining induction step allows us to reduce the size of the last non-empty trapezoid twice. For an illustration see Figure 8. Lemma 4.11 (Bisection of a trapezoid). Let L > 0. Let r be an
ad-missible sequence such that for some i ∈ [2k] we have ri+1 = 0 and ri
larger than some sufficiently large constant. Letλ = min{ℓ > 0, ℓui+1∈ Z2} = O(1) and let x = ui+1λ⌊ri/(2λhui+k, ui+1i)⌋. With this choice hui+k, xi ≃ ri/2. Then we set:
er = (r0, . . . , ri−1, hui+k, xi, 0, . . . , 0),
¯r = (r0− huk, xi, . . . , ri− huk+i, xi, 0, . . . , 0), ¯ L = min L, L − hu0, xi + huk, xi hu0, u2k−1i hu2k−1, uki , e V = VL(er), V = x + VL¯(¯r).
With these notations,
γV 6γVe/µ(GV) + γV/µ(GVe).
Proof of Lemma 4.11. We begin with some geometric observations,
which follow directly from Definitions 3.4 and 3.5.
Claim 4.12. The sequenceser, r are admissible. Furthermore, we have e
V ∪ V = V and V \ eV = T i.
Proof of the claim. The admissibility ofer follows from that of r, since e
ri < ri and eri+1 = 0. For r notice that huj, ui+1i > 0 for all i ∈ [2k] andj ∈ [k, k + i] with equality iff i = 2k − 1 and j = k. Thus, for all j we have rj − rj > 0 and these differences are within a factor O(1) of each other (except forr0− r0 if it is0) and similarly for L − L0. In particular, admissibility is preserved, sincerj > 0 for all j 6 i (by the previous observation it suffices to check this forj = i).
x
e
V
V
i−1T
iFIGURE 8. The geometric setting of Lemma 4.11. The
two hatched snails are eV and V . The dotted-hatched region isVi−1while the dotted trapezoid is Ti.
By Definition 3.5 it is clear thatV = eV ⊔ T with T = Hk+i(Rk+i+ ri) \ Hk+i(Rk+i+ huk+i, xi)
∩ Hk+i−1(Rk+i−1+ ri−1) ∩ Hk+i+1(Rk+i+1). It also follows from Definition 3.5 thatT = Tias claimed. Finally, we have thatV ⊂ V by Definition 3.4, which completes the proof.
Let now
Vi−1= VL(r0, . . . , ri−1, 0, . . . , 0) = V \ Ti = V ∩ eV . By Claim 4.12 we have
(ΩV, µV(· | GV)) = (ΩVe, µVe(· | GVe)) ⊗ (ΩTi, µTi(· | GTi)) (16) and we can apply Lemma 3.10 with the conditioning event
GVi−1 = GT ∩\ j<i GT¯j ⊂ ΩVe. We get VarV(f | GV) 6 µ(GVi−1) −1µ V VarVe(f | GVe) +1{G V i−1}VarTi(f | GTi) | GV , (17) where we used once more the Harris inequality to bound from below µVe(GVi−1| GVe) by µ(GVi−1). Using the definition of the Poincar´e con-stant γVe, the fact that c
e V
x 6 cVx together withµ(GVi−1) > µ(GV) the first term is bounded from above by
γVe µ(GV)µV X x∈ eV cVx Varx(f ) . (18)
The second term, µ(GVi−1)−1µ V 1{G
V i−1}VarTi(f | GTi) | GV
, in the r.h.s. of (17) can be bounded from above by
µ(GVe)−1µVe µVi−1(VarTi(f | GTi) | GVi−1) 6µ(GVe) −1µ e V VarV(f | GV) 6 γV µ(GV) µV X x∈V cVx Varx(f ) , (19)
using (16), Lemma 3.9, the definition ofγV and the fact thatcV
x 6cVx. If we now combine together (17), (18) and (19) we get the
state-ment of the lemma.
We can now assemble our main induction step from Lemmas 4.10 and 4.11.
Corollary 4.13 (Removing a trapezoid). With the notation of Lemma
4.11 and the additional hypothesisri−1>2q−2w we have γV 6 Lq−w 4 /(min ˆ V µ(GVˆ))log ri O(1) max ˆ Vi−1 γVˆi−1,
where the minVˆ runs over all (admissible) snails VLˆ(ˆr) such that 0 6 rj − ˆrj = O(ri) for all j 6 i, 0 6 L − ˆL = O(ri) and ri+1 = 0 and maxVˆi−1 runs over those additionally satisfyingrˆi = 0.
Proof of Corollary 4.13. We apply Lemma 4.11 to V and then repeat-edly to the resulting snails V and eV . Note that ri + eri = ri and e
ri/(λhui+k, ui+1i) = ⌊ri/(λhui+k, ui+1i/2⌋, so in log ri+ O(1) iterations we reach snails withri = λhui+k, ui+1i = O(1), to which Lemma 4.10 can be applied. Let us begin by checking that the hypotheses of the two lemmas do remain valid. By Claim 4.12 all sequences appearing are admissible and smaller than L, r. Furthermore, from the defini-tion of ri−1 it is clear that all values of ri−1 obtained in the process areri−1− O(ri) > 1/(qwδ), so indeed Lemma 4.10 applies to the final snails with aO(1) lines in their i-th trapezoid. Similarly, values of rj andL change by O(ri). In particular, the final trapezoids are indeed covered bymaxVˆi−1 and intermediate ones – by minVˆ.
The desired result then follows directly from Propositions 4.10 and
4.11 as described above.
We are now ready to conclude the proof of Proposition 4.9. Indeed, applying Corollary 4.13 once for each non-zero coordinate of r, we obtain γV 6γT ew 3 / min ˆ V µ(GVˆ) O(log L)
5. PROOF OF THEOREM 1
5.1. Roadmap. Before turning to the details of the proof of Theo-rem 1, let us sketch our approach.
We aim to prove that the infection time is at most eO(log(1/q)3)/qα w.h.p., so up to this time scale it is harmless to consider the KCM process restricted to the torus Λ centered at the origin with side ew3log(1/q)3/qα
. Next, we define an event E ⊂ ΩΛ∩Z2 in such a way that, roughly, the following holds:
(i) in any snail V ⊂ Λ of size Θ(1/q3w) the good events G± i (cf. Definition 4.2) occur for all i ∈ [2k].
(ii) in each strip of Λ parallel to u0 and of width 2R there exists a tube T (cf. Definition 3.5) of length 1/q3w and radius R such that the good eventGT holds.
It is then easy to verify thatµ(E) > 1 −e−q−Θ(w)
and conclude that it is sufficient to analyse the infection time of the origin of the equilibrium U-KCM chain in Λ restricted to the E. For the latter chain we follow the standard “variational” route (see [15, Section 2.2]) and get that the mean infection time is at most
q−1sup f
VarΛ(f | E) DΛ(f )
,
where the supremum is taken over all f : E → R. The last and most important step is to prove that
VarΛ(f | E) 6 eO(w(log(1/q))
3)/qα
DΛ(f ). (20)
The intuition behind the proof of (20) is as follows. The “infected” snails, namely those for which GT occurs, move in Λ mimicking a generalized 1-neighbour model with an effective density (see [15, Section 3.1])
qeff = µ(GT | E) = eΘ(log(1/q))
2/qα .
Thanks to Theorem 4.5, each step of the generalized 1-neighbour model requires a time eO(w(log(1/q))3)/qα
. Indeed, the snail (see Fig-ure 6) does extend on both sides of the tube for a distance Θ(L), so one can induce a change on both sides of the tube by resampling the configuration inside the snail. The conclusion of Theorem 1 then follows rather naturally.
Λ(2)1 Λ(1)1 W = 1 q3w W + 2R0 2R0≈2w 2 qα log 1 q
FIGURE 9. The partition of Λ into strips Λi = Λ(1)i ∪ Λ(2)i , i ∈ [M]. The hatched region represents a square Qi,j.
5.2. Proof. Let
K = 2 exp(w3(log q)3/qα), wherew is the usual large constant, and let
Λ = R2/(Ku0Z+ KukZ) be the torus in R2 of side K directed by u
0, which we think of as centred at0. Further set
R =w2log(1/q)/qα W =1/q3w M =K/(2R0+ W ), recalling the notationR0 = ρ0⌊R/ρ0⌋ from Definition 3.3. For simplic-ity we assume thatu0(2R0 + W ) ∈ Z2 and that M is an even integer (W and K can be modified by O(1) and O(1/q3w) respectively, so that these both hold).
We partition Λ into alternating strips Λ(1)i , Λ(2)i , i ∈ [M], of length K and parallel to u0 (see Figure 9). The strips {Λ(1)i }i∈[M ] have all width 2R0 while the strips {Λ(2)i }i∈[M ] have width W . We write Λi = Λ(1)i ∪ Λ(2)i and we think of the thin strip Λ(1)i as being just below the thick one Λ(2)i , when u0 points left. In turn, we partition Λi into consecutive squaresQi,j, j ∈ [M], of side length equal to 2R0+ W and sides parallel tou0 anduk and we writeQ(a)i,j = Qi,j ∩ Λ(a)i , a = 1, 2.
Remark 5.1. Recalling Definition 3.3, the width of the thin strips is chosen so that a ringR of radius R would fit tightly inside.
Now let t = K/w and let τ0, τ0Λ denote the infection times of the origin for theU-KCM process on Z2 and for theU-KCM process on the discrete torusΛ ∩ Z2 respectively. Using the boundedness of the jump
propagation (see e.g. [13]) implies that
Pµ τ0 > t 6 PµΛ τ0Λ> t+ e−Ω(K) asq → 0. (21) Given a small positive constantε, let
(i) Ai,j be the event that the rightmost and leftmost ringsR in Q(1)i,j are infected and any segmentI ⊂ Q(1)i,j intersecting Z2, of length εR and orthogonal to ui for somei ∈ [−k + 1, k − 1] contains an infectedu-helping set in Q(1)i,j;
(ii) Ei,j be the event that any segment I ⊂ Qi,j intersecting Z2, of length εW and orthogonal to some u ∈ ˆS contains w infected consecutive sites.
With our choice of K, R, W it follows (see e.g. [15, Lemma 6.5]) thatµ(Ai,j) = qO(Rw) and
µ \
j∈[M ] Ei,j
>1 − O MW2)e−qwεW/w >1 − e−ε/(2wq2w)).
Then, forλ = O(1) to be specified later, the Harris inequality gives
µ \
j∈[λ] Ai,j
>qO(Rw). (22)
In conclusion, if Ei denotes the event that for all the squares Qi,j in Λi the event Ei,j holds and moreover there exists j ∈ [M] such that the eventTj+λj′=j+1Ai,j′ also holds, then
µ(Ei) > 1 − e−Ω(1/q
2w) .
Therefore, letting E = Ti∈[M ]Ei and writing τAΛ for the hitting time of A ⊂ {0, 1}Λ for the U-KCM process in Λ, recalling that t = K/w, we obtain Pµ Λ(τ Λ Ec 6t) 6 X i∈[M ] Pµ Λ(τ Λ Ec i 6t) 6 X i∈[M ] O(KW t)µ(Eic) + e−Ω(KW t)= e−ε/(3wq2w). (23) In the second inequality above we used a simple union bound over the updates for theU-KCM in Λ together with the fact that the law of the U-KCM process in Λ started from µΛ is equal toµΛ at any given time and a simple large deviations result on the number of updates.
Thus, ifF = {ω : ω0 = 0} ∪ Ec then (21) together with (23) imply that Pµ(τ0 >t) 6 PµΛ(τ Λ F >t) + e−ε/(4wq 2w) asq → 0. (24) As in [15] Pµ Λ(τ Λ F >t) 6 e−λFt, (25) with λF = inf DperΛ (f ) µΛ(f2) , f |F = 0 ,
where DΛper(f ) denotes the Dirichlet form of the U-KCM process on the torusΛ (see (8)). Observe now that for any f : ΩΛ → R such that f |F = 0 VarΛ(f | E) = 1 2 X ω X ω′ µΛ(ω | E)µΛ(ω′| E)(f (ω) − f (ω′))2 >µΛ(ω0 = 0 | E)µΛ(f2| E) > qµΛ(f2),
where for the last inequality we used the Harris inequality ({ω : ω0 = 0} and E are both decreasing events) and the fact that f2
1{E} = f 2. Hence, λF >q inf f DperΛ (f ) VarΛ(f | E) .
Notice the absence of the conditioning event E in the Dirichlet form DperΛ (f ). Our main result on the above variational problem is as fol-lows.
Theorem 5.2. For all w > 0 large enough, all ε > 0 small enough and
allf : ΩΛ → R VarΛ(f | E) 6 eO(w(log q) 3)/qα) DΛper(f ), (26) i.e. λF >e−O(w(log q) 3)/qα) .
Before proving (26) let us first complete the proof of Theorem 1.
Proof of Theorem 1. Usingt = K/w = w−1exp(w3(log q)3/qα), we get that for anyw large enough
tλF ≫ 1/q2w asq → 0, which, together with (24) and (25), gives
Eµ(τ0) = Z +∞ 0 ds Pµ(τ0 > s) 6t + Z T t ds Pµ(τ0 > s) + Z +∞ T ds Pµ(τ0 > s).
Taking w > 3α, the second term in the r.h.s. above tends to zero as q → 0 by (27). The last one also tends to zero because Pµ(τ0 > s) 6 e−sλ0 with (see [15, Theorem 2])λ
0 >e−Ω((log q)
4/q2α)
.
Proof of Theorem 5.2. The two main ingredients of the proof will be
Lemma 3.12 and Theorem 4.5. Using µΛ(· | E) = NiµΛi(· | Ei) and Lemma 3.9, we first write
VarΛ(f | E) 6 X
i
µΛ(VarΛi(f | Ei) | E). (28) Hence, it is enough to analyse a generic term µΛ(VarΛi(f | Ei) | E) and for this purpose we apply Lemma 3.12 with parameters ˆS ∼ Sj = ΩQi,j, νj = µQi,j(· | Ei,j), n = M, k = λ and constraining event A ∼ Aj = Ai,j ⊂ Sj to the product space. Indeed, the lemma applies by (22). Thus, writingκA
i,j for the indicator of j+λ\ j′=j+1 Ai,j′∪ j−1\ j′=j−λ Ai,j′, we get VarΛi(f | Ei) 6 q −O(Rwλ2)X j µΛi(κ A
i,jVarQi,j(f | Ei,j) | Ei), which, combined with (28), implies that
VarΛ(f | E) 6 q−O(Rwλ
2)X
i,j
µΛ(κAi,jVarQi,j(f | Ei,j) | E).
We shall now analyse a generic term µΛ(κAi,jVarQi,j(f | Ei,j) | E) with the help of Theorem 4.5 to prove the following.
Lemma 5.3. For any functionf : ΩΛ→ R µΛ(κAi,jVarQi,j(f | Ei,j) | E) 6 q
−O(Rw log(W ))X x∈Λ d(x,Qi,j)6O(λW ) µΛ cxVarx(f ) .
If we assume the lemma we immediately recover (26), concluding
Proof of Lemma 5.3. We assume that κA
i,j = 1 and, w.l.o.g., that A+ =
j+λ\ j′=j+1
Ai,j′
occurs. Next, we recall Definition 3.4 of the snailVL(r) and we choose rl = ρk+l⌊δrl−1/ρk+l⌋ with r−1 = L = (λ−1)W with λ sufficiently large depending on δ and U, but not on w or q, so that Qi,j ⊂ x + VL(r), where x is the center of the rightmost ring in Q(1)i,j+λ. We denoteV = x + VL(r). By construction, Qi,j∩ T = ∅. Finally, we recall Definition 4.2 of the good eventsG0, G±i andG = GT ∩Ti∈[2k](G+i ∩ G−i ) for the snailV and we observe that
1{A+}(ω) 6 1{G T}, (29) 1{E} 61 {Ti(G + i ∩G − i)}. (30)
Let nowa := µV(f | G). Using µΛ(E) = 1 − o(1) we have that µΛ(1A+VarQ i,j(f | Ei,j) | E) 6(1 + o(1))µΛ µQi,j(1{G T}1{E}(f − a) 2) 6(1 + o(1))µΛ µQi,j(1{G}(f − a) 2) 6µΛ µV(1{G}(f − a) 2)/µ(G) = µΛ VarV(f | G) ,
using that µ(E) = 1 − o(1), (29) and the fact that the expectation minimises the L2 distance among constants for the first inequality and (30) for the second one. If we now apply the bound (12) of Theorem 4.5 and use the fact that V is contained in a deterministic O(λW )-neighborhood of the square Qi,j we get the conclusion of the lemma, noting thatcV
x 6cx, wherecx are the constraints on the torus
Λ.
6. OPEN PROBLEMS
With Theorem 1 establishing universality, the next natural goal is to determine the relaxation time up to a constant factor. This would correspond to reaching the refined universality partition in bootstrap percolation proved in [4]. However, for KCM, we expect that the partition, even restricted to the finite stable set case studied in this work, will be more subtle.
stable directions and difficultyα. Then Eµ[τ0] = exp
logγ(1/q)(log log(1/q))O(1) qα
,
where
• γ = 0 if there exists at most one direction u ∈ S1 such that α(u) > α (balanced unrooted models),
• γ = 1 if there exist at least two directions u ∈ S1, such that α(u) > α, but not two opposite ones (balanced rooted models), • γ = 2 if there exists u such that min(α(u), α(u + π)) > α, but
α(v) 6 α for v 6= u, u + π (unbalanced unrooted models), • γ = 3 otherwise, i.e. there exist three different directions with
difficulty larger than α, two of which are opposite (unbalanced
rooted models).
Furthermore, we conjecture that the (log log(1/q))O(1) correction is in
factΘ(1), except, possibly, for models with exactly one direction u ∈ S1
withα(u) > α.
It should be noted that such sharp results are not known for any critical model, so Conjecture 6.1 provides the highest precision cur-rently feasible. In fact, the level of precision of the conjecture is not attained for any model, including the most classical FA2f model falling in the first class.
Out of the four cases, the lower bounds for the first and third ones follow directly from bootstrap percolation results [4, Theorem 1.4] and [16, Lemma 4.3]. The upper bound in the fourth case is Theo-rem 2. The Theo-remaining upper bounds are work in progress and require finer mechanisms for moving droplets on scales smaller than the ones considered here. The simplest case,γ = 2, is [15, Conjecture 4]. For the caseγ = 1 see Remark 4.6. The remaining lower bounds for the second and fourth cases are likely to be hard. For the corresponding conjecture for critical models with infinitely many stable directions see [11, Conjecture 7.1].
APPENDIX
Proof of Lemma 3.12. We will consider the non-periodic case - the
pe-riodic one is treated identically. For simplicity we assume that2k di-videsn. Partition [n] into blocks I0, . . . , IN −1 whereIi := {ik, . . . , (i + 1)k − 1} and N = n/k. Let κ(ℓ)(ω) be the indicator of the event that there exists i ∈ [N](ℓ) := [N/2] such that ω
j ∈ A for all j ∈ Ii. Let κ(r)(ω) be defined similarly but for the blocks with index in [N](r) :=
[N] \ [N](ℓ). Using the assumption of the lemmamax(ν(1 − κ(ℓ)), ν(1 − κ(r))) < 1/16 and [3, Lemma 6.5] gives that
Varν(f | ΩA) 6 24 ν κ(r)Var(ℓ)(f ) + κ(ℓ)Var(r)(f ) | ΩA
,
where Var(ℓ) denotes the variance computed w.r.t. the product mea-sureNj<kN/2νj and similarly forVar(r).
Given κ(ℓ)(ω) = 1, let ξ be the smallest label in [N](ℓ) such that ωj ∈ A ∀j ∈ Iξ. Using Lemma 3.9 and the fact that {ω : ξ = i} is independent of the variables{ωj}j>ik, we get that
ν κ(ℓ)Var(r)(f ) | ΩA
6 X
i∈[N ](ℓ)
ν(1{ξ=i}Var{>ik}(f ) | ΩA), (31)
whereVar{>ik}(f ) is the variance computed w.r.t. Nj>ik νj. The latter can now be bounded above using [15, Proposition 3.4]. If κ±j is the indicator of the eventωi ∈ A for all i ∈ Ij±1 with the convention that κ+N −1= 0 we get that
1{ξ=i}Var{>ik}(f ) 6 ˆν(A)
−O(k) 1{ξ=i} N −1X j=i+1 ν{>ik} κjVarIj(f ) , where κj = 1 − (1 − κ−j )(1 − κ+j ) and VarIj is the variance w.r.t. the variables inIj. By inserting the r.h.s. above into the r.h.s. of (31), we get thatν κ(ℓ)Var(r)(f ) | Ω
A is smaller than ˆ ν(A)−O(k) N −1X j=1 ν νIj−1∪Ij+1(κjVarIj(f )) j−2 X i=0 1ξ=i + N/2−1 X i=0
ν 1ξ=iν{>ik}(κi+1VarI
i+1(f )) 62ˆν(A)−O(k) N −1X j=1 ν κjVarIj(f ) ,
isolating the termj = i+1 and usingPi1ξ=i 61 and1ξ=i 61 for the
two terms respectively. Exactly the same argument can be applied to the termν κ(r)Var(ℓ)(f )to conclude that
Varν(f | ΩA) 6 96ˆν(A)−O(k) N X j=1 ν (κ+j + κ−j ) VarIj(f ) . (32)
We finally bound from above a generic term, ν κ±j VarIj(f )
, consid-ering w.l.o.g. κ+1.
1} and conditioning event E = {ω : ωk−1 ∈ A} to get VarI1(f ) 6 2ˆν(A) −1ν I1 Vark(f ) +1{ω k∈A}VarI1\{k}(f ) . (33)
If we insert the r.h.s. above into the r.h.s. of (32), we get the termsν κ+1 Varνk(f ) = ν κA k Varνk(f )) and ν κ + 11{ω k∈A}VarI1\{k}(f ) weighted by192ˆν(A)−O(k)+1. Notice that
κ+11{ω
k∈A} 6κ
A k−1
and therefore we can repeat the above step for the set I1 \ {k − 1} with Ω1 = {k − 2}, Ω2 = [k − 2] and E = {ωk−2 ∈ A}. At the end of the iteration we finally get that
ν κ+1 VarI1(f ) 6ν(A)ˆ −O(k2)X i∈I1 ν κAi Varνi(f ) . Putting all together we have finally proved that
Varν(f ) 6 (2/ˆν(A))O(k) N X j=1 ν (κ+ j + κ−j) VarIj(f ) 6(2/ν(A))O(k)X j X i∈Ij ν κAi Varνi(f ) . ACKNOWLEDGEMENTS
We wish to thank Laure Marˆech´e for many enlightening discussions concerning universality forU-KCM.
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E-mail address: hartarsky@ceremade.dauphine.fr
CEREMADE, CNRS, UMR 7534, UNIVERSIT´E PARIS-DAUPHINE, PSL UNIVER
-SITY, PLACE DUMAR´ECHAL DELATTRE DETASSIGNY, 75775 PARISCEDEX16, FRANCE
E-mail address: martin@mat.uniroma3.it
DIPARTIMENTO DI MATEMATICA E FISICA, UNIVERSIT`A ROMA TRE, LARGO S.L. MURIALDO, 00146, ROMA, ITALY
E-mail address: toninelli@ceremade.dauphine.fr
CEREMADE, CNRS, UMR 7534, UNIVERSIT´E PARIS-DAUPHINE, PSL UNIVER
-SITY, PLACE DUMAR´ECHAL DELATTRE DETASSIGNY, 75775 PARISCEDEX16, FRANCE
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