Universality for critical KCM: infinite number of stable directions
aa r X i v : . [ m a t h . P R ] A p r Universality for critical kinetically constrained models:infinite number of stable directions
Ivailo Hartarsky ∗ ,1,2 , Laure Marêché † ,3 , and Cristina Toninelli ‡ ,2 DMA UMR 8553, École Normale SupérieureCNRS, PSL Research University45 rue d’Ulm, 75005 Paris, France CEREMADE UMR 7534, Université Paris-DauphineCNRS, PSL Research UniversityPlace du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France LPSM UMR 8001, Université Paris DiderotCNRS, Sorbonne Paris Cité75013 Paris, France
April 22, 2019
Abstract
Kinetically constrained models (KCM) are reversible interacting particle systemson Z d with continuous-time constrained Glauber dynamics. They are a natural non-monotone stochastic version of the family of cellular automata with random initial stateknown as U -bootstrap percolation. KCM have an interest in their own right, owing totheir use for modelling the liquid-glass transition in condensed matter physics.In two dimensions there are three classes of models with qualitatively different scal-ing of the infection time of the origin as the density of infected sites vanishes. Here westudy in full generality the class termed ‘critical’. Together with the companion paperby Martinelli and two of the authors [20] we establish the universality classes of criticalKCM and determine within each class the critical exponent of the infection time as wellas of the spectral gap. In this work we prove that for critical models with an infinitenumber of stable directions this exponent is twice the one of their bootstrap percolationcounterpart. This is due to the occurrence of ‘energy barriers’, which determine thedominant behaviour for these KCM but which do not matter for the monotone boot-strap dynamics. Our result confirms the conjecture of Martinelli, Morris and the lastauthor [26], who proved a matching upper bound. MSC2010:
Primary 60K35; Secondary 82C22, 60J27, 60C05
Keywords:
Kinetically constrained models, bootstrap percolation, universality, Glauberdynamics, spectral gap. ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Kinetically constrained models (KCM) are interacting particle systems on the integer lat-tice Z d , which were introduced in the physics literature in the 1980s by Fredrickson andAndersen [16] in order to model the liquid-glass transition (see e.g. [17, 31] for reviews), amajor and still largely open problem in condensed matter physics [5]. A generic KCM is acontinuous-time Markov process of Glauber type characterised by a finite collection U of fi-nite nonempty subsets of Z d zt u , its update family . A configuration ω is defined by assigningto each site x P Z d an occupation variable ω x P t , u , corresponding to an empty or occupied site respectively. Each site x P Z d waits an independent, mean one, exponential time andthen, iff there exists U P U such that ω y “ for all y P U ` x , site x is updated to empty withprobability q and to occupied with probability ´ q . Since each U P U is contained in Z d zt u ,the constraint to allow the update does not depend on the state of the to-be-updated site.As a consequence, the dynamics satisfies detailed balance w.r.t. the product Bernoulli( ´ q )measure, µ , which is therefore a reversible invariant measure. Hence the process started at µ is stationary.Both from a physical and from a mathematical point of view, a central issue for KCMis to determine the speed of divergence of the characteristic time scales when q Ñ . Twokey quantities are: (i) the relaxation time T rel , i.e. the inverse of the spectral gap of theMarkov generator (see Definition 2.5) and (ii) the mean infection time E p τ q , i.e. the meanover the stationary process started at µ of the first time at which the origin becomes empty.Several works have been devoted to the study of these time scales for some specific choicesof the constraints [2, 9, 12, 13, 25, 27] (see also [17] section 1.4.1 for a non exhaustive list ofreferences in the physics literature). These results show that KCM exhibit a very large varietyof possible scalings depending on the update family U . A question that naturally emerges,and that has been first addressed in [26], is whether it is possible to group all possible updatefamilies into distinct universality classes so that all models of the same class display the samedivergence of the time scales.Before presenting the results and the conjectures of [26], we should describe the keyconnection of KCM with a class of discrete monotone cellular automata known as U -bootstrappercolation (or simply bootstrap percolation) [8]. For U -bootstrap percolation on Z d , givenan update family U and a set A t of sites infected at time t , the infected sites in A t remaininfected at time t ` , and every site x becomes infected at time t ` if the translate by x of one of the sets in U is contained in A t . The set of initial infections A is chosen atrandom with respect to the product Bernoulli measure with parameter q P r , s , whichidentifies with µ : for every x P Z d we have µ p x P A q “ q . One then defines the criticalprobability q c ` Z d , U ˘ to be the infimum of the q such that with probability one the wholelattice is eventually infected, namely Ť t ě A t “ Z d . A key time scale for this dynamics isthe first time at which the origin is infected, τ BP . In order to study this infection time formodels on Z , the update families were classified by Bollobás, Smith and Uzzell [8] into threeuniversality classes: supercritical , critical and subcritical , according to a simple geometriccriterion (see Definition 2.1). In [8] they proved that q c ` Z , U ˘ “ if U is supercritical orcritical, and it was proved by Balister, Bollobás, Przykucki and Smith [4] that q c ` Z , U ˘ ą if U is subcritical. For supercritical update families, [8] proved that τ BP “ q ´ Θ p q w.h.p.as q Ñ , while in the critical case τ BP “ exp p q ´ Θ p q q . The result for critical families waslater improved by Bollobás, Duminil-Copin, Morris and Smith [7], who identified the criticalexponent α “ α p U q such that τ BP “ exp p q ´ α ` o p q q .Back to KCM, if we fix an update family U and an initial configuration ω and we identify2he empty sites with infected sites, a first basic observation is that the clusters of sites thatwill never be infected in the U -bootstrap percolation correspond to clusters of sites which areoccupied and will never be emptied under the KCM dynamics. A natural issue is whetherthere is a direct connection between the infection mechanism of bootstrap percolation andthe relaxation mechanism for KCM, and, more precisely, whether the scaling of T rel and E p τ q is connected to the typical value of τ BP when the law of the initial infections is µ . It is notdifficult to establish that µ p τ BP q provides a lower bound for E p τ q and T rel (see [27, Lemma 4.3]and (10)), but in general, as we will explain, this lower bound does not provide the correctbehaviour.In [26], Martinelli, Morris and the last author proposed that the supercritical class shouldbe refined into unrooted supercritical and rooted supercritical models in order to capture thericher behavior of KCM. For unrooted models the scaling is of the same type as for bootstrappercolation, T rel „ E p τ q “ q ´ Θ p q as q Ñ [26, Theorem 1(a)] , while for rooted modelsthe divergence is much faster, E p τ q „ T rel “ e Θ pp log q q q (see [26, Theorem 1(b)] for the upperbound and [25, Theorem 4.2] for the lower bound).Concerning the critical class, the lower bound with µ p τ BP q mentioned above and the re-sults of [8] on bootstrap percolation imply that T rel and E p τ q diverge at least as exp p q ´ Θ p q q .In [26, Theorem 2] an upper bound of the same form was established and a conjecture [26,Conjecture 3] was put forward on the value of the critical exponent ν such that both E p τ q and T rel scale as exp p| log q | O p q { q ν q , with ν in general different from the exponent of thecorresponding bootstrap percolation process. Furthermore, a toolbox was developed for thestudy of the upper bounds, leading to upper bounds matching this conjecture for all mod-els. The main issue left open in [26] was to develop tools to establish sharp lower bounds.A first step in this direction was done by Martinelli and the last two authors [25] by an-alyzing a specific critical model known as the Duarte model for which the update familycontains all the -elements subsets of the North, South and West neighbours of the origin.Theorem 5.1 of [25] establishes a sharp lower bound on the infection and relaxation timesfor the Duarte KCM that, together with the upper bound in [26, Theorem 2(a)], proves E Duarte p τ q “ exp p Θ pp log q q { q qq as q Ñ , and the same result holds for T rel . The divergenceis again much faster than for the corresponding bootstrap percolation model, for which itholds τ BP “ e Θ pp log q q { q q w.h.p as q Ñ [30] (see also [6], from which the sharp value ofthe constant follows), namely the critical exponent for the Duarte KCM is twice the criticalexponent for the Duarte bootstrap percolation.Both for Duarte and for supercritical rooted models, the sharper divergence of time scalesfor KCM is due to the fact that the infection time of KCM is not well approximated bythe infection mechanism of the monotone bootstrap percolation process, but is instead theresult of a much more complex infection/healing mechanism. Indeed, visiting regions of theconfiguration space with an anomalous amount of empty sites is heavily penalised and requiresa very long time to actually take place. The basic underlying idea is that the dominantrelaxation mechanism is an East-like dynamics for large droplets of empty sites. Here East-like means that the presence of an empty droplet allows to empty (or fill) another adjacentdroplet but only in a certain direction (or more precisely in a limited cone of directions). Thisis reminiscent of the relaxation mechanism for the East model, a prototype one-dimensionalKCM for which x can be updated iff x ´ is empty, thus a single empty site allows tocreate/destroy an empty site only on its right (see [15] for a review on the East model). Forsupercritical rooted models, the empty droplets that play the role of the single empty sites For the lower bound of T rel one does not need to use the boostrap percolation results, as T rel ě q ´ min U P U | U | {| U | by plugging the test function t ω “ u in Definition 2.5. q eff “ q Θ p q . Forthe Duarte model, droplets have a size that diverges as ℓ “ | log q |{ q and thus an equilibriumdensity q eff “ q ℓ “ e ´p log q q { q . Then a (very) rough understanding of the results of [25, 26] isobtained by replacing q with q eff in the time scale for the East model T East rel “ e Θ pp log q q q [2]. Themain technical difficulty to translate this intuition into a lower bound is that the dropletscannot be identified with a rigid structure. In [25] this difficulty for the Duarte modelwas overcome by an algorithmic construction that allows to sequentially scan the system insearch of sets of empty sites that could (without violating the constraint) empty a certainrigid structure. These are the droplets that play the role of the empty sites for the Eastdynamics.In [26] all critical models which have an infinite number of stable directions (see Sec-tion 2.1), of which the Duarte model is but one example, were conjectured to have a criticalexponent ν “ α , with α “ α p U q the critical exponent of the corresponding bootstrap per-colation dynamics (defined in Definition 2.2). The heuristics is the same as for the Duartemodel, the only difference being that droplets would have in general size ℓ “ | log q | O p q { q α .However, the technique developed in [25] for the Duarte model relies heavily on the specificform of the Duarte constraint and in particular on its oriented nature , and it cannot beextended readily to this larger class.In this work, together with the companion paper by Martinelli and two of the authors [20],we establish in full generality the universality classes for critical KCM, determining the criticalexponent for each class.Here we treat all choices of U for which there is an infinite number of stable directions and prove (Theorem 2.8) a lower bound for T rel and E p τ q that, together with the matchingupper bound of [26, Theorem 2], yields E p τ q “ e | log q | O p q { q α for q Ñ and the same result for T rel . Our technique is somewhat inspired by the algorithmicconstruction of [25], however, the nature of the droplets which move in an East-like way ishere much more subtle, and in order to identify them we construct an algorithm which canbe seen as a significant improvement on the α -covering and u -iceberg algorithms developedin the context of bootstrap percolation [7].In the companion paper [20] we prove for the complementary class of models, namely allcritical models with a finite number of stable directions , an upper bound that (together withthe lower bound from bootstrap percolation) yields instead E p τ q “ e | log q | O p q { q α for q Ñ and the same result for T rel .A comparison of our results with Conjecture 3 of [26] is due. The class that we considerhere is, in the notation of [26], the class of models with bilateral difficulty β “ 8 , hence belongto the α -rooted class defined therein. Therefore, our Theorem 2.8 proves Conjecture 3(a) inthis case. We underline that it is not a limitation of our lower bound strategy that prevents usfrom proving Conjecture 3(a) for the other α -rooted models, namely those with α ď β ă 8 .Indeed, as it is proven in the companion paper [20], in this case the conjecture of [26] is notcorrect, since it did not take into account a subtle relaxation mechanism which allows torecover the same critical exponent as for the bootstrap percolation dynamics. Note that, since the Duarte update rules contain only the North, South and West neighbours of theorigin, the constraint at a site x does not depend on the sites with abscissa larger than the abscissa of x . Before turning to our models of interest, KCM, let us recall recent universality results forthe intimately connected bootstrap percolation models in two dimensions. U -bootstrap per-colation (or simply bootstrap percolation) is a very general class of monotone transitive localcellular automata on Z first studied in full generality by Bollobás, Smith and Uzzell [8]. Let U , called update family , be a finite family of finite nonempty subsets, called update rules ,of Z zt u . Let A , called the set of initial infections , be an arbitrary subset of Z . Thenthe U -bootstrap percolation dynamics is the discrete time deterministic growth of infectiondefined by A “ A and, for each t P N , A t ` “ A t Y t x P Z : D U P U , U ` x Ă A t u . In other words, at any step each site becomes infected if a rule translated at it is already fullyinfected, and infections never heal. We define the closure of the set A by r A s “ Ť t ě A t andwe say that A is stable when r A s “ A . The set of initial infections A is chosen at randomwith respect to the product Bernoulli measure µ with parameter q P r , s : for every x P Z we have µ p x P A q “ q .Arguably, the most natural quantity to consider for these models is the typical (e.g. mean)value of τ BP , the infection time of the origin.The combined results of Bollobás, Smith and Uzzell [8] and Balister, Bollobás, Przykuckiand Smith [4] yield a pre-universality partition of all update families into three classes withqualitatively different scalings of the median of the infection time as q Ñ . In order to definethis partition we will need a few definitions.For any unitary vector u P S “ t z P R : } z } “ u ( } ¨ } denotes the Euclidean normin R ) and any vector x P R we denote H u p x q “ t y P R : x u, y ´ x y ă u — the openhalf-plane directed by u passing through x . We also set H u “ H u p q . We say that a direction u P S is unstable (for an update family U ) if there exists U P U such that U Ă H u and stable otherwise. The partition is then as follows. Definition 2.1 (Definition 1.3 of [8]) . An update family U is • supercritical if there exists an open semi-circle of unstable directions, • critical if it is not supercritical, but there exists an open semi-circle with a finite numberof stable directions, • subcritical otherwise. 5he main result of [8] then states that in the supercritical case τ BP “ q ´ Θ p q with highprobability as q Ñ , while in the critical one τ BP “ exp p q ´ Θ p q q . The final justification ofthe partition in Definition 2.1 was given by Balister, Bollobás, Przykucki and Smith [4] whoproved that the origin is never infected with positive probability for subcritical models for q ą sufficiently small, i.e. q c ` Z , U ˘ ą if U is subcritical. From the bootstrap percolationperspective supercritical models are rather simple, while subcritical ones remain very poorlyunderstood (see [19]). Nevertheless, most of the non-trivial models considered before theintroduction of U -bootstrap percolation, including the -neighbour model (see [1, 22] forfurther results), fall into the critical class, which is also the focus of our work.Significantly improving the result of [8], Bollobás, Duminil-Copin, Morris and Smith [7]found the correct exponent determining the scaling of τ BP for critical families. Moreover,they were able to find log τ BP up to a constant factor. To state their results we need thefollowing crucial notion. Definition 2.2 (Definition 1.2 of [7]) . Let U be an update family and u P S be a direction.Then the difficulty of u , α p u q , is defined as follows. • If u is unstable, then α p u q “ . • If u is an isolated stable direction (isolated in the topological sense), then α p u q “ min t n P N : D K Ă Z , | K | “ n, |r Z X p H u Y K qsz H u | “ 8u , (1)i.e. the minimal number of infections allowing H u to grow infinitely. • Otherwise, α p u q “ 8 .We define the difficulty of U by α p U q “ inf C P C sup u P C α p u q , (2)where C “ t H u X S : u P S u is the set of open semi-circles of S .It is not hard to see (Theorem 1.10 of [8], Lemma 2.6 of [7]) that the set of stabledirections is a finite union of closed intervals of S and that (Lemmas 2.7 and 2.10 of [7]) (1)also holds for unstable and strongly stable directions, that is directions in the interior of theset of stable directions (but not for semi-isolated stable directions i.e. endpoints of non-trivialstable intervals). Furthermore (see [7, Lemma 2.7], [8, Lemma 5.2]), ď α p u q ă 8 if andonly if u is an isolated stable direction, so that U is critical if and only if ď α p U q ă 8 . Asa final remark we recall that, contrary to determining whether an update family is critical,finding α p U q is a NP-hard question [21].We are now ready to describe the universality results. A weaker form of the result of [7]is that τ BP “ exp p q ´ α p U q` o p q q with high probability as q Ñ . For the full result however, weneed one last definition. Definition 2.3.
A critical update family U is balanced if there exists a closed semi-circle C such that max u P C α p u q “ α p U q and unbalanced otherwise.Then [7] provides that for balanced models τ BP “ exp p Θ p q{ q α p U q q with high probabilityas q Ñ , while for unbalanced ones τ BP “ exp p Θ pp log q q q{ q α p U q q . These are the best generalestimates currently known. We refer to [28, 29] for recent surveys on these results as well ason sharper results for some specific models. 6 .2 Kinetically constrained models Returning to KCM, let us first define the general class of KCM introduced by Cancrini,Martinelli, Roberto and the last author [9] directly on Z . Fix a parameter q P r , s and anupdate family U as in the previous section. The corresponding KCM is a continuous-timeMarkov process on Ω “ t , u Z which can be informally defined as follows. A configuration ω is defined by assigning to each site x P Z an occupation variable ω x P t , u correspondingto an empty (or infected ) and occupied (or healthy ) site respectively. Each site waits anindependent exponentially distributed time with mean before attempting to update itsoccupation variable. At that time, if the configuration is completely empty on at least oneupdate rule translated at x , i.e. if D U P U such that ω y “ for all y P U ` x , then we performa legal update or legal spin flip by setting ω x to with probability q and to with probability ´ q . Otherwise the update is discarded. Since the constraint to allow the update neverdepends on the state of the to-be-updated site, the product measure µ is a reversible invariantmeasure and the process started at µ is stationary. More formally, the KCM is the Markovprocess on Ω with generator L acting on local functions f : Ω ÞÑ R as p L f qp ω q “ ÿ x P Z c x p ω q p µ x p f q ´ f q p ω q , (3)where µ x p f q denotes the average of f with respect to the variable ω x conditionally on t ω y u y ‰ x ,and c x is the indicator function of the event that there exists U P U such that U ` x iscompletely empty, i.e. ω U ` x ” . We refer the reader to chapter I of [24], where the generaltheory of interacting particle systems is detailed, for a precise construction of the Markovprocess and the proof that L is the generator of a reversible Markov process t ω p t qu t ě on Ω with reversible measure µ .The corresponding Dirichlet form is defined as D p f q “ ÿ x P Z µ ` c x Var x p f q ˘ , (4)where Var x p f q denotes the variance of the local function f with respect to the variable ω x conditionally on t ω y u y ‰ x . The expectation with respect to the stationary process with initialdistribution µ will be denoted by E “ E q, U µ . Finally, given a configuration ω P Ω and a site x P Z , we will denote by ω x the configuration obtained from ω by flipping site x , namelyby setting p ω x q x “ ´ ω x and p ω x q y “ ω y for all y ‰ x . For future use we also need thefollowing definition of legal paths, that are essentially sequences of configurations obtainedby successive legal updates. Definition 2.4 (Legal path) . Fix an update family U , then a legal path γ in Ω is a finitesequence γ “ ` ω p q , . . . , ω p k q ˘ such that, for each i P t , . . . , k u , the configurations ω p i ´ q and ω p i q differ by a legal (with respect to the choice of U ) spin flip at some vertex v “ v p ω p i ´ q , ω p i q q .As mentioned in Section 1, our goal is to prove sharp bounds on the characteristic timescales of critical KCM. Let us start by defining precisely these time scales, namely the re-laxation time T rel (or inverse of the spectral gap) and the mean infection time E p τ q (withrespect to the stationary process). Definition 2.5 (Relaxation time T rel ) . Given an update family U and q P r , s , we say that C ą is a Poincaré constant for the corresponding KCM if, for all local functions f , we haveVar µ p f q “ µ p f q ´ µ p f q ď C D p f q . (5)7f there exists a finite Poincaré constant, we define T rel “ T rel p q, U q “ inf t C ą C is a Poincaré constant u . Otherwise we say that the relaxation time is infinite.A finite relaxation time implies that the reversible measure µ is mixing for the semigroup P t “ e t L with exponentially decaying time auto-correlations (see e.g. [3, Section 2.1]). Definition 2.6 (Infection time τ ) . The random time τ at which the origin is first infectedis given by τ “ inf t ě ω p t q “ ( , where we adopt the usual notation letting ω p t q be the value of the configuration ω p t q at theorigin, namely ω p t q “ p ω p t qq . The East model
We close this section by defining a specific example of KCM on Z , theEast model of Jäckle and Eisinger [23], which will be crucial to understand our results (KCMon Z are defined in the same way as KCM on Z ). It is defined by an update family composedby a single rule containing only the site to the left of the origin ( ´ ). In other words, site x can be updated iff x ´ is empty. For this model both T rel and E p τ q scale as exp ´ p log q q ¯ as q Ñ [2, 9, 12] . One of the key ingredients behind this scaling is the following combinatorialresult [32] (see [14, Fact 1] for a more mathematical formulation). Proposition 2.7.
Consider the East model on t , . . . , M u defined by fixing ω “ at all time.Then any legal path γ connecting the fully occupied configuration (namely ω s.t. ω x “ forall x P t , . . . , M u ) to a configuration ω such that ω M “ goes through a configuration withat least r log p M ` q s empty sites. This logarithmic ‘energy barrier’, to employ the physics jargon, and the fact that atequilibrium the typical distance to the first empty site is M “ Θ p { q q are responsible for thedivergence of the time scales as roughly { q r log p M ` q s “ e Θ pp log q q q . In this paper we study critical KCM with an infinite number of stable directions or, equiva-lently, with a non-trivial interval of stable directions.
Theorem 2.8.
Let U be a critical update family with an infinite number of stable directions.Then there exists a sufficiently large constant C ą such that E p τ q ě exp ` { ` Cq α p U q ˘˘ , as q Ñ and the same asymptotics holds for T rel . This theorem combined with the upper bound of Martinelli, Morris and the last author [26,Theorem 2(a)], determines the critical exponent of these models to be α in the sense ofCorollary 2.9 below. We thus complete the proof of universality and Conjecture 3(a) of [26]for these models . Actually these references focus on the study of T rel . A matching upper bound for E p τ q follows from (10).The lower bound for E p τ q follows easily from the lower bound for P p τ ą t q with t “ exp p log p q q { q obtained in the proof of Theorem 5.1 of [11]. The conjecture involuntarily asks for a positive power of log q , which we do not expect to be systematicallypresent (see Conjecture 7.1). orollary 2.9. Let U be a critical update family with an infinite number of stable directions.Then q α p U q log E p τ q “ p´ log q q O p q as q Ñ and the same holds for T rel . Universality for the remaining critical models is proved in a companion paper by Martinelliand the first and third authors [20] and, in particular, Conjecture 3(a) of [26] is disprovedfor models other than those covered by Theorem 2.8. It is important to note that Theo-rem 2.8 significantly improves the best known results for all models with the exception ofthe recent result of Martinelli and the last two authors [25] for the Duarte model. Indeed,the previous bound had exponent α , and was proved via the general (but in this case farfrom optimal) lower bound with the mean infection time for the corresponding bootstrappercolation model [27, Lemma 4.3]. In this section we outline roughly the strategy to derive our main result, Theorem 2.8.The hypothesis of infinite number of stable directions provides us with an interval of stabledirections. We can then construct stable ‘droplets’ of shape as in Figure 3 (see Definitions 5.2and 5.3), where we recall from Section 2.1 that a set is stable if it coincides with its closure.Thus, if all infections are initially inside a droplet, this will be true at any time under the KCMdynamics. The relevance and advantage of such shapes come from the fact that only infectionssituated to the left of a droplet can induce growth left. This is manifestly not feasible withoutthe hypothesis of having an interval of stable directions. It is worth noting that these shapes,which may seem strange at first sight, are actually very natural and intrinsically present in thedynamics. Indeed, such is the shape of the stable sets for a representative model of this class– the modified 2-neighbour model with one (any) rule removed, that is the three-rule updatefamily with rules tp´ , q , p , qu , tp´ , q , p , ´ qu , tp , ´ q , p , qu (it can also be seen as themodified Duarte model with an additional rule). The stable sets in this case are actuallyYoung diagrams.We construct a collection of such droplets covering the initial configuration of infections,so that it gives an upper bound on the closure. To do this, we devise an improvement ofthe α -covering algorithm of Bollobás, Duminil-Copin, Morris and Smith [7]. It is importantfor us not to overestimate the closure as brutally. Indeed, a key step and the main difficultyof our work is the Closure Proposition 5.17, which roughly states that the collections ofdroplets associated to the closure of the initial infections is equal to the collection for theinitial infections. This is highly non-trivial, as in order not to overshoot in defining thedroplets, one is forced to ignore small patches of infections (larger than the ones in [7]),which can possibly grow significantly when we take the closure for the bootstrap percolationprocess and especially so if they are close to a large infected droplet. In order to remedy thisproblem, we introduce a relatively intrinsic notion of ‘crumb’ (see Definition 5.1) such thatits closure remains one and does not differ too much from it. A further advantage of ouralgorithm for creating the droplets over the one of [7] is that it is somewhat canonical, with awell-defined unique output, which has particularly nice ‘algebraic’ description and properties(see Remark 5.7). Another notable difficulty we face is systematically working in roughlya half-plane (see Remark 5.18 for generalisations) with a fully infected boundary condition,but we manage to extend our reasoning to this setting very coherently.9 u v v u ` πu ´ π Figure 1: Illustration of Lemma 4.1 and its proof. Thickened arcsrepresent intervals of strongly stable directions. Solid dots repre-sent isolated and semi-isolated stable directions. The difficulties ofthe isolated stable directions are indicated next to them and yieldthat the difficulty of the model is α “ . The directions chosen inLemma 4.1 are the solid vectors u , u , v “ v and a direction v in the strongly stable interval ending at v sufficiently close to v .Note that the definition of v (and v ) disregards stable directionswith difficulty smaller than α as present on the figure.Finally, having established the Closure Proposition 5.17 alongside standard and straight-forward results like an Aizenmann-Lebowitz Lemma 5.10 and an exponential decay of theprobability of occurrence of large droplets (Lemma 5.12), we finish the proof via the follow-ing approach, inspired by the one developed by Martinelli and the last two authors [25] forthe Duarte model. The key step here (see Section 6) is mapping the KCM legal paths tothose of an East dynamics via a suitable renormalisation. Roughly speaking, we say that arenormalised site is empty if it contains a large droplet of infections. However, for the renor-malised configuration to be mostly invariant under the original KCM dynamics, we ratherlook for the droplets in the closure of the original set of infections instead. This is where theClosure Proposition 5.17 is used to compensate the fact that the closure of equilibrium is notequilibrium. In turn, this mapping together with the combinatorial result for the East modelrecalled in Section 2.2 (Proposition 2.7), yield a bottleneck for our dynamics correspondingto the creation of log p { q eff q droplets, where { q eff is the equilibrium distance between twoempty sites in the renormalized lattice, and q eff „ e ´ { q α . This provides for the time scalesthe desired lower bound q eff log p q eff q „ e { q α of Theorem 2.8. The last part of the proof fol-lows very closely the ideas put forward in [25] for the Duarte model. However, in [25], therewas no need to develop a subtle droplet algorithm since, owing to the oriented character ofthe Duarte constraint, droplets could simply be identified with some large infected verticalsegments. It is also worth noting that, thanks to the less rigid notion of droplets that wedevelop in the general setting, some of the difficulties faced in [25] for Duarte are no longerpresent here. Let us fix a critical update family U with an infinite number of stable directions for the restof the paper. We will omit U from all notation, such as α p U q .The next lemma establishes that one can make a suitable choice of stable directions,which we will use for all our droplets. At this point the statement should look very odd andtechnical, but it simply reflects the fact that we have a lot of freedom for the choice and wemake one which will simplify a few of the more technical points in later stages. Nevertheless,this is to a large extent not needed besides for concision and clarity.A direction u P S is called rational if tan u P Q Y t8u . Lemma 4.1.
There exists rational stable directions S “ t u , u , v , v u (see Figure 1) withdifficulty at least α such that • The directions appear in couterclockwise order u , u , v , v . • No u P S is a semi-isolated stable direction. u ´ i belongs to the cone spanned by v i and u i for i P t , u i.e. the strictly smallerinterval among r v i , u i s and r u i , v i s contains u ´ i . • is contained in the interior of the convex envelope of S . • Either u ă v ´ π { or u ą v ` π { . • p H u Y H u q X Z is stable or, equivalently, E U P U , U Ă H u Y H u . • the directions u “p u ` u q{ ,u “p u ` u q{ ,u “p u ` u q{ are rational.Proof. Since U has an infinite number of stable directions and they form a finite union ofclosed intervals with rational endpoints [8, Theorem 1.10], there exists a non-empty openinterval I of stable directions. Further note that the set J of directions u such that thereexists a rule U P U and x P U with x x, u y “ is finite, so one can find a non-trivial closedsubinterval I Ă I which does not intersect J . The directions u and u will be chosen in I ,which clearly implies that they are strongly stable and thus with infinite difficulty. Moreover,if there exists U P U with U Ă H u Y H u , by stability of u , we have U X p H u z H u q ‰ ∅ ,which contradicts I X J “ ∅ .Since U is critical it does not have two opposite strongly stable directions, so there isno strongly stable direction in I ` π . If there are any (isolated or semi-isolated) stabledirections in I ` π , we can further choose a non-trivial open subinterval I Ă I , for whichthis is not the case (there is a finite number of isolated and semi-isolated stable directions).Let π ą δ ą be such that the angle between any two consecutive directions of difficultyat least α is at most π ´ δ (it is well defined by (2)). We then choose a non-trivial closedsubinterval I Ą I “ r u , u s with u rational and u “ p u ` u q{ rational and with ă u ´ u ă δ ă π . It easily follows from the sum and difference formulas for the tangentfunction that u , u and u are also rational.Let v “ max t v P p u , u ` π q : α p v q ě α u ,v “ min t v P p u ´ π, u q : α p v q ě α u . These both exist, since I ` π does not contain stable directions, both p u , u ` π q and p u ´ π, u q contain directions with difficulty at least α by (2) and the set of such directions isclosed. If v is not semi-isolated, we set v “ v and similarly for v . Otherwise, we choose arational strongly stable direction sufficiently close to v as v and similarly for v . We claimthat this choice satisfies all the desired conditions. Indeed, all directions in S are stable non-semi-isolated rational with difficulty at least α and the last but one condition was alreadyverified.One does have that u is in the cone spanned by v and u , which is implied by v Pp u ´ π, u q and similarly for u , so the third condition is also verified. If v ´ v ě π , thenthere is an open half circle contained in p v , v q with no direction of difficulty at least α ,which contradicts (2), so v ´ v ă π and the same holds for u ´ v , u ´ u and v ´ u by11 B Λ u u u Figure 2: The open domain B defined in (6) is shaded, while its complement Λ is not. The lines are the boundaries of the three half-planes defining B . Notethat if a R H u , then Λ becomes simply a cone.the definition of v and v , the fact that v and v are sufficiently close to them and the factthat I was chosen smaller than π . Thus is in the convex envelope of S .Finally, if one has both v ´ u ď π { and u ´ v ď π { , then one obtains v ´ v ą π ´ δ ,since I is smaller than δ . However, v and v are consecutive directions of difficulty at least α , which contradicts the definition of δ .For the rest of the paper we fix directions S “ t u , u , v , v u as in Lemma 4.1 and assumewithout loss of generality that u ă v ´ π { .Let us fix large constants ! C ! C ! C ! C ! C ! C ! C , each of which can depend on previous ones as well as on U and S . We will also use asymptoticnotation whose constants can depend on U and S , but not on C or the other constants above.All asymptotic notation is with respect to q Ñ , so we assume throughout that q ą issufficiently small.For any two sets K, B Ă R we define r K s B “ rp K Y Bq X Z szB .Finally, we make the convention that throughout the article all distances, balls and di-ameters are Euclidean unless otherwise stated. We say that a set X Ă R is within distance δ of a set Y Ă R if d p x, Y q ď δ for all x P X where d is the Euclidean distance. In this section we define our main tool – the droplet algorithm. It can be seen as a significantimprovement on the α -covering and u -iceberg algorithms [7, Definitions 6.6 and 6.22], manyof whose techniques we adapt to our setting.We will work in an infinite domain Λ defined as follows (see Figure 2). Fix some vector a P R and let B “ H u Y H u p a q Y H u p a q , Λ “ R zB (6)where the directions u , u and u are those defined in Lemma 4.1. In other words, Λ is acone with sides perpendicular to u and u cut along a line perpendicular to u . The readeris invited to simply think that B is a half-plane directed by u , which will not change thereasoning. 12 .1 Clusters and crumbs Let Γ be the graph with vertex set Z but with x „ y if and only if } x ´ y } ď C . Let Γ bedefined similarly with C replaced by C . Definition 5.1 (Clusters and crumbs) . Fix a finite set K Ă Λ X Z of infected sites. Let G Ă K be a connected component of the subgraph of Γ induced by K . Then G is a crumb ifit is at distance more than C from B and there exists a set P G Ă Z such that r P G s Ą G and | P G | “ α ´ . Let κ Ă K be a connected component which is not a crumb. We call cluster any C Ă κ such that the induced subgraph of Γ is connected and diam p C q ď C and suchthat C is maximal with this property. We call boundary cluster every cluster at distance atmost C from B .We similarly define modified crumb , modified cluster and modified boundary cluster byreplacing Γ and C by Γ and C respectively.Clearly, any (modified) non-boundary cluster has at least α sites. Indeed, if its connectedcomponent is of diameter larger than C , then the diameter of the cluster is larger than C ´ C , and we can choose C large enough to get C ´ C C ě α , while otherwise the clusteris a connected component which is not a crumb and at distance more than C from B , so bydefinition has at least α sites. Moreover, a cluster only intersects a bounded number of otherclusters, as its diameter is bounded. Also note that crumbs (resp. modified crumbs) are atdistance at least C (resp. C ) from any other site of K Y B and have diameter much smallerthan C , as we shall see in Corollary 5.14. The proofs of this corollary and Observation 5.13 itfollows from are both independent of the rest of the argument and postponed for convenience,but we allow ourselves to use this (easy) result ahead of these proofs.Let C be a cluster (resp. modified cluster). We denote by Q p C q (resp. Q p C q ) the smallestopen quadrilateral with sides perpendicular to S containing the set t x P R : d p x, C q ă C u (resp. C ). Note that Q p C q Ą r C s (resp. Q p C q ), since Q p C q X Z Ą C (resp. Q p C q ) isstable and that diam p Q p C qq “ Θ p C q (resp. diam p Q p C qq “ Θ p C q ), as diam p C q ď C . Weextend the definition Q p C q for (non-modified) clusters. We now define the shape that our ‘droplets’ will have, which resembles Young diagrams .The following definitions are illustrated in Figure 3. Definition 5.2 (DYD) . We call distorted Young diagram (DYD) a subset of R of the form p H v p x q X H v p x qq X č i P I p H u p x i q Y H u p x i qq (7)for a finite set I , some set X “ t x i : i P I u of vectors x i P R and x P R . The vectors x i and x are uniquely defined up to redundancy (and up to the convention that all x i are on thetopological boundary of the DYD). An alternative definition of the DYD can also be givenas p H v p x q X H v p x qq X ď i P I p H u p y i q X H u p y i qq , (8)where y i are the convex corners of the diagram rather than the concave ones. For the 3-rule model alluded to in Section 3 stable sets consist precisely of Young diagrams and thedirections S provided by Lemma 4.1 can be arbitrarily close to the four axis directions, yielding Youngdiagrams. x x x x xy y y y y v v u u u D B Figure 3: The shaded region D is adistorted Young diagram (DYD) asin Definition 5.2. The larger quadri-lateral with vertices x , x , y and x is Q p D q . Note that Q p D q can degen-erate into a triangle, but we call ita quadrilateral nevertheless. On thefigure | D | is the length of the v side,but this is not always the case. Thethickened region is the cut distortedYoung diagram (CDYD) C p D q of D .The vertical line is the boundary be-tween Λ on its left and B on its right.For any DYD D we denote by y the vector such that x y, u j y “ sup a P D x a, u j y “ max i P I x y i , u j y for j P t , u . We further denote Q p D q “ H u p y q X H u p y q X H v p x q X H v p x q , i.e. the minimal quadrilateral containing D with sides directed by S . In these terms, Q (resp. Q ) is a DYD and Q p Q q “ Q . Definition 5.3 (CDYD) . We call cut distorted Young diagram (CDYD) a subset of R ofthe form Λ X p H u p y q X H u p y qq X č i P I p H u p x i q Y H u p x i qq for a finite set I and some vectors x i P R and y P Λ . Alternatively, one can write Λ X ď i P I p H u p y i q X H u p y i qq , where y i P Λ are the convex corners.For a DYD, D , we define C p D q as the CDYD defined by the same x i and y or the same y i . We extend the notation C p D q to CDYD by setting C p D q “ D if D is a CDYD. Notethat by Lemma 4.1 all DYD and CDYD are stable for the bootstrap percolation dynamics(restricted to Λ ). Also pay attention to the fact that CDYD are not necessarily connected,contrary to DYD. 14 efinition 5.4 (Size) . For a DYD D we set π p D q “ t x P R : D y P D, x y, v ` π { y “ x u tobe its projection (parallel to v ) and | D | “ sup π p D q ´ inf π p D q to be its size – the length ofthe projection. For a CDYD D we denote its size | D | “ diam p D q{ C .Note that if D is a DYD, then | D | “ | Q p D q| by Lemma 4.1 and the assumption we madethat u ă v ´ π { . Furthermore, for all DYD diam p D q “ Θ p| D |q again by Lemma 4.1with constants depending only on S . One should be careful with the meaning of size fordisconnected CDYD, but it will not cause problems, as all CDYD arising in our forthcomingalgorithm are connected. Observation 5.5.
Note that for any d ě the number of discretised DYD and CDYD (i.e.intersections of a DYD or CDYD with Z ) containing a fixed point a P R of diameter atmost d is less than c d for some constant c depending only on S . Proof.
Note that a DYD or CDYD is uniquely determined by its rugged edge formed byits u and u -sides. However, this edge injectively defines an oriented percolation path withdirections perpendicular to u and u on the lattice t x P R : D x , x P Z , x x, u y “ x x , u y , x x, u y “ x x , u yu (except its endpoints, which lie on similar lattices). Since the graph-length of this path isbounded by O p d q and its endpoints are within distance d from a , the result follows. We next introduce a procedure of merging DYD and CDYD. This will be used only forcouples of intersecting ones, but can be defined regardless of whether they intersect. Theoperation is illustrated in Figure 4.
Lemma 5.6.
For any two DYD, D and D , the minimal DYD containing D Y D is welldefined. We denote it by D _ D and call it their span . The operation _ is associative andcommutative.Proof. Let D be defined by Y “ t y i : i P I u , x (see (8)) and similarly for D . Let x P R bethe vector such that H v i p x q Y H v i p x q “ H v i p x q for i P t , u . Let Y be the set of y i P Y Y Y such that for all y j P Y Y Y with y i ‰ y j we have H u p y j q X H u p y j q Č H u p y i q X H u p y i q .We denote by D the DYD defined by Y, x and claim that for any DYD D Ą D Y D we have D Ą D , which is enough to conclude that D “ D _ D is well defined. Let D be definedby Y , x .Note that for each y i P Y (and in fact in Y Y Y ) there is a sequence of points in D or D converging to y i , so that (by extraction of a subsequence) there exists y j with H u p y j q X H u p y j q Ą H u p y i q X H u p y i q . Similarly, there is a sequence of points in D or D converging to the boundary of H v p x q , so that H v p x q Ą H v p x q and similarly for v . Thus,we do have D Ą D .Finally, the commutativity is obvious and the associativity follows from the characterisa-tion of D _ D as the minimal DYD containing both D and D .We analogously define the span D _ D of two CDYD D and D – the minimal CDYDcontaining both – and note that it coincides with their union (which is also commutative and Associativity was referred to as commutativity by previous authors [8]. x x x “ x y y “ y y “ y y “ y y x y “ y y “ y y “ y y “ y x x x “ x x “ x y x xy x x x x x D D D _ D Figure 4: The shaded region D and thickened region D are DYD. Their respective quadrilat-erals Q p D i q are completed by dashed lines. Their span D _ D is hatched and its quadrilateral Q p D _ D q is also completed by dashed lines.associative). We also define the span C _ D of a DYD D and a CDYD C as the minimalCDYD containing p C Y D qzB , which coincides with C _ C p D q . The proof that it is welldefined is analogous to Lemma 5.6.We have thus defined an associative and commutative binary operation _ on all DYDand CDYD. Moreover, the idempotent unary operation C p¨q is distributive with respect to _ and C p D q _ D “ C p D _ D q . Furthermore, the span of several DYD is the minimalDYD containing all of them, while the span of several DYD and at least one CDYD is theminimal CDYD containing all the corresponding CDYD. We call droplet a DYD or CDYD included in Λ . We are now ready to define our dropletalgorithm (resp. modified droplet algorithm ), which takes as input a finite set K Ă Λ X Z ofinfections and outputs a set D of disjoint connected droplets. It proceeds as follows. • Form an initial collection of DYD D consisting of Q p C q (resp. Q p C q ) for all clusters16resp. modified clusters) C of K . If a DYD D P D intersects B , replace it by its CDYD, C p D q , to obtain a droplet. • As long as it is possible, replace two intersecting droplets of D by their span. If thespan intersects B , replace it by its CDYD to obtain a droplet. • Output the collection D obtained when all droplets are disjoint.The output D is clearly a collection of disjoint connected droplets. Indeed, by inductionall x i corners of droplets remain in Λ (see Figure 4), so that DYD remain connected whenreplaced by CDYD. Remark 5.7.
From the results of Section 5.3 it is clear that the order of merging does notimpact the output of the algorithm, which is thus well defined. It can also be expressed asthe minimal collection of disjoint droplets containing the intersection with Λ of the originalcollection of quadrilaterals. This minimal collection is well defined. Consequently, the unionof the output is increasing in the input. Definition 5.8 (Spanned droplets) . Let D be a droplet and K be a finite set. We say that D is spanned (resp. modified spanned ) for K with boundary B if the output of the dropletalgorithm (resp. modified droplet algorithm) for K X D has a droplet containing D . We omit K and B if they are clear from the context.Note that, when seen as an event, a droplet being spanned is monotone, contrary to whatis the case in [7, 8], which formally invalidates the proofs therein. It is also clear that eachdroplet appearing in (the intermediate or final stages of) the droplet algorithm is spannedand similarly for the modified droplet algorithm. Indeed, the clusters responsible for creatinga droplet in the course of the algorithm are contained in the droplet, so each of them is stilla cluster of K X D (recall that crumbs have diameter much smaller than C ). We next establish several properties of the algorithm. The approach is similar to the oneof [7] with the notable exception of the key Closure Proposition 5.17. We start with thefollowing purely geometric statement.
Lemma 5.9 (Subadditivity) . Let D and D be two DYD or CDYD with non-empty inter-section. Then | D _ D | ď | D | ` | D | . Furthermore, if D is a DYD intersecting B , then | C p D q| ď | D | .Proof. First assume that D and D are DYD. Since | D | “ | Q p D q| for any DYD D and D _ D Ă Q p Q p D q _ Q p D qq , it suffices to prove the assertion for merging quadrilateralsinstead of DYD. But in that case it is not hard to check directly and is a particular case ofLemma 15 of the first arXiv version of [8] (or Lemma 23 of the second version). Since similar(but actually slightly more involved) details were omitted in the proof of the correspondingLemma 4.6 of [8] and differed to earlier versions, we will not go into useless detail here either.To give a sketch of a possible argument, one can check that for fixed shapes of Q p D q and Q p D q the maximal Q p Q p D q _ Q p D qq is achieved when their intersection is reduced to avertex. Yet, in those configurations one can obtain the v and v sides of Q p Q p D q _ Q p D qq
17s the union of those of Q p D q and translates of those of Q p D q (see Figure 4). This concludesthe proof, as only v and (possibly) v sides contribute to | ¨ | by Lemma 4.1.Next assume that D is a DYD and D is a CDYD. Let Y “ t y i : i P I u be the set ofvectors defining C p D q and let a P D X D . Since Y Ă D , we have that d p y i , a q ď diam p D q .It then easily follows that the CDYD defined by only one corner, y i , which we denote C p y i q ,is within distance O p diam p D qq from C p a q . But then C p D q “ Ť i P I C p y i q is within distance O p diam p D qq from C p a q . Thus, | D _ D | ď p diam p D q ` O p diam p D qqq{ C ď | D | ` | D | ,since diam p D q “ O p| D |q and all implicit constants depend only on S and are thus muchsmaller than C .Next assume that D and D are CDYD. Then the statement is trivial, because D _ D “ D Y D , so diam p D q ` diam p D q ě diam p D _ D q by the triangle inequality.Finally, let D be a DYD intersecting B . Then, | C p Q p D qq| ě | C p D q| and | Q p D q| “ | D | ,so we may assume that D “ Q p D q and prove | C p D q| ď | D | . But in this case it is easy tosee that diam p C p D qq “ O p diam p D qq “ O p| D |q with constants depending only on S , whichconcludes the proof.The subadditivity lemma will be used to prove the next two adaptations of classicalresults. Lemma 5.10 (Aizenman-Lebowitz) . Let K be a finite set and let D be a spanned (resp.modified spanned) droplet with | D | ě C . Then for all C { C ď k ď | D |{ C there exists aconnected spanned (resp. modified spanned) droplet D with k ď | D | ď k .Proof. By Lemma 5.9 at each step of the droplet algorithm (resp. modified droplet algorithm)the largest size of a droplet appearing in the collection at most doubles. Initially the largestsize is at most C C and in the end there is a (unique) droplet D Ą D , so that | D | ě| D |{ C ě C { C ą C C . Then there is a stage of the algorithm at which the maximalsize of a droplet in D is between k and k , which is enough since all droplets appearing inthe droplet algorithm (resp. modified droplet algorithm) are connected and spanned (resp.modified spanned). Lemma 5.11 (Extremal) . Let K be a finite set and let D be a spanned droplet. Then thetotal number of disjoint clusters in D is at least diam p D q{ C .Proof. Assume that at the initial stage of the algorithm there are k clusters (not disjoint).One can then find k { C disjoint ones, since their diameter is at most C . Yet at each step ofthe algorithm the number of CDYD plus twice the number of DYD decreases by at least , sothat there are at most k ´ steps. Furthermore, by Lemma 5.9 the total size of droplets inthe collection D is decreasing, so that | D |{ C ď | D | ď kC C , where D Ą D is some dropletin the output of the algorithm. Indeed, | Q p C q| ď C C for all clusters C . This concludes theproof, since | D | ě diam p D q{ C for all DYD and CDYD.We next transform this extremal bound into an exponential decay of the probability thata droplet is spanned until saturation at the critical size. Lemma 5.12 (Exponential decay) . Let D be a droplet with | D | ď {p C q α q . Then µ p D is spanned q ă exp p´ C | D |q . Proof.
Let D be a droplet with | D | ď {p C q α q , so that diam p D q “ d ď C {p C q α q . ByLemma 5.11 if D is spanned, it contains at least d { C disjoint clusters, each one having18iameter at most C . Each non-boundary cluster has at least α sites, while boundary clustersare non-empty and located at distance at most C from B . Thus, we have the union bound µ p D is spanned q ď d { C ÿ l “ ˆ C α d l ˙ˆ C dd { C ´ l ˙ q lα `p d { C ´ l q ď d { C ÿ l “ d {p C q p C q α d { l q l .e d ` d { C ÿ l “ d {p C q p C qd { l q l .e d ď exp p´ C d q . Our next aim is to prove that the closure of a set is contained in its droplet collectionup to very local infections next to initial ones. To that end we will need some preliminaryresults, similar to those used by Bollobás, Duminil-Copin, Morris and Smith [7].
Observation 5.13 (Lemma 6.5 of [7]) . Let u be a rational non-semi-isolated stable direction.Let K Ă Z with | K | ă α p u q (if α p u q “ 8 the condition is that K is finite, but there is noa priori bound on its size). Then there exists a constant C p U , u, | K |q not depending on K such that r K s H u is within distance C p U , u, | K |q from K .Since we will require some improvements later, we spell out a proof of the above resultfor completeness (actually our proof is slightly different from the one in [7]). Proof of Observation 5.13.
We prove the statement by induction on | K | . For a K “ t x u thisis easy, since if x x, u y is sufficiently large r K s H u “ K and otherwise there is a single possibleconfiguration for each value of x x, u y up to translation. Assume the result holds for | K | ă n .If one can write K “ K \ K with K , K ‰ ∅ and d p K , K q ą C p U , u, n ´ q ` O p q , then r K s H u “ r K s H u \ r K s H u , since r K s H u and r K s H u are at sufficiently large distance, henceno site can use both to become infected. Assume that, on the contrary, there are no largegaps between parts of K . There is a finite number of such K up to translation and for eachof these r K s is finite (e.g. since K is contained in a quadrilateral with sides perpendicularto S ), so within uniformly bounded distance from K . Therefore, if H u is sufficiently farfrom K , r K s H u “ r K s . Otherwise, there is a finite number of possible K up to translationperpendicular to u and for each of them r K s H u is finite, so that one can indeed find a finiteuniform constant C p U , u, n q as claimed.A quantitative version of this result was proved by Mezei and the first author [21]. Aneasy corollary of Observation 5.13 is the fact that crumbs can only grow very locally (seeFigure 5a). Corollary 5.14.
Let C be sufficiently large depending on U . Let K Ă Z with | K | ă α .Then r K s is within distance C {p α q from K . Also, for a (modified) crumb G we have that diam pr G sq ď αC and r G s is within distance C from G .Proof. The first assertion follows from Observation 5.13, since if it were wrong, one couldsimply translate a set K sufficiently far from a half-plane yielding a contradiction with theobservation.Next consider a (modified) crumb G and P G minimal with | P G | ă α and r P G s Ą G . Then r G s Ă r P G s is within distance C {p α q from P G . If the sites of P G are not connected in thegraph Γ on Z with connections at distance at most C ` C , then either G is not connectedin Γ or P G is not minimal, which are both contradictions. Similarly, if there is no site of G at19 ď C (a) The dots represent thesites of a crumb. The (discon-nected) circled shape boundsits closure. Note that crumbsmay have gaps of size C whilethe growth allowed is only C ! C . y y x xy y C u { C C v { C C D D (b) The shaded region is the shrunken DYD D of the largest DYD D . The solid circles represent crumbs and the dashed arcs are thebound for their growth provided by Lemma 5.16. The modifiedclusters of the closure are included in the dotted DYD. Figure 5: Illustrations of Corollary 5.14, Lemma 5.16 and Proposition 5.17.distance smaller than C {p α q from a C {p α q -connected component of P G , that componentcan be removed from P G , contradicting minimality. Hence, P G is within distance C { from G . The result is then immediate, as r G s is within distance C { ` C {p α q from G and itsdiameter is at most C {p α q ` diam p P G q , while diam p P G q ď p α ´ qp C ` C q .In order to treat infection at the concave corners of droplets we will need the followingmodification of Observation 5.13. Corollary 5.15.
Let u and u be rational strongly stable directions such that H u Y H u isstable for the bootstrap percolation dynamics i.e. E U P U , U Ă H u Y H u . Let K Ă Z with | K | ď α ´ . Then r K s H u Y H u is within distance C p U , u , u q from K .Proof. We apply a similar induction to the one in the proof of Observation 5.13. The onlydifference is that we can no longer use translation invariance. If d p K, H u q ą C p U , u , | K |q ` O p q , by Observation 5.13, we have r K s H u Y H u “ r K s H u and similarly for u and u inter-changed. We can thus assume that K is within distance C p U , u , u q from the origin. Butthen r K Y H u Y H u s Ă H u Y H u Y H u p C p U , u , u q u q , where u “ p u ` u q{ , since thelatter region is stable by the hypothesis on u , u .We next transform these results for infinite regions into a result for droplets. It statesthat a crumb next to a droplet cannot grow significantly (see Figure 5b). Lemma 5.16.
Let C be sufficiently large depending on U and S . Let D be a DYD atdistance at least C from B or be a CDYD and let G be a crumb. Then r G s D YB “ r G s D iswithin distance C of G . roof. Assume that D is a DYD at distance at least C from B . The proof of [7, Lemma 6.10]applies using (7), Observation 5.13, Corollary 5.15 and the arguments in the proof of Corollary5.14 to give the result for r G s D , which is therefore at distance at least C ´ C from B since d p G, Bq ě C , so that in fact r G s D “ r G s D YB .Assume next that D is a CDYD. Then actually D Y B can be viewed as a DYD on theentire plane without boundary specified by an infinite number of vectors x i , so that we arein the previous case. In order to avoid introducing the corresponding notion of infinite DYD,one can consider an increasing exhaustive sequence of DYD D i converging to D Y B in theproduct topology and apply the previous result for r G s D i , which will thereby apply to D Y B .Finally, r G s D “ r G s D YB follows, since d pr G s D YB , Bq ě C ´ C .The next proposition is key to making the output of the algorithm essentially invariantunder the KCM dynamics without having to pay for the fact that the closure for the bootstrappercolation dynamics of infections at equilibrium is not at all at equilibrium itself. The proofis illustrated in Figure 5b. Proposition 5.17 (Closure) . Let K be a finite set and D be the collection of droplets givenby the modified droplet algorithm with input r K s B . Let D be the output of the droplet algorithmfor K . Then @ D P D D D P D , D Ă D. Proof.
Let G be the set of crumbs for K . Set G “ Ť G P G G . Claim 1.
For each crumb G P G its closure r G s “ r G s B consists of at most α ´ modifiedcrumbs all contained within distance C from G . Proof of Claim 1.
There exists a set P G as in Definition 5.1, such that r P G s Ą G and thus r P G s Ą r G s , which proves that all connected components of r G s for Γ are modified crumbs.The fact that r G s is within distance C of G (and thus at distance at least C from B ) wasproved in Corollary 5.14, which also shows that r G s “ r G s B , since G is at distance more than C from B .We can thus define G p G q to be the set of modified crumbs of r G s B , so that their unionis disjoint and equal to r G s B . Moreover, crumbs in G are at distance at least C from eachother, so for any two of them G ‰ G we have that any G P G p G q and G P G p G q are atdistance at least C ´ C " C and also at such distance from B , so that r G s B “ Ť G P G r G s B has no modified cluster and consists of modified crumbs at distance at most C from G .For a droplet D P D consider the set of vectors Y and x ( x is absent for CDYD) definingit. Then define Y “ Y ` C u { C and x “ x ` C v { C , where u P R is the vector suchthat x u , u y “ x u , u y “ ´ and v is defined identically in terms of v and v . We denote D the droplet defined by Y and x and call it shrunken droplet . Let D “ Ť D P D D and D “ Ť D P D D . It is clear that D is at distance at least C { C from Λ z D for all droplets D . In particular, all shrunken droplets are at distance at least C { C from each other andshrunken DYD are at distance at least C { C from B , so that Lemma 5.16 applies to themand r D s B “ D . Claim 2. D Y G Ą K . Proof of Claim 2.
Note that it is enough to prove that the clusters of K are contained in D .Assume that there exists a P K z D and a P C for some cluster. Then, Q p C q X Λ is containedin some D P D , which is defined by Y and x ( x is absent for CDYD). Then since a R D ,21ither for all y i P Y we have a R H u p y i q X H u p y i q or a R H v p x q X H v p x q . In the former case, a ´ C u { C R H u p y i q X H u p y i q for all y i P Y . However, Q p C q contains the ball of radius C centered at a and } u } “ O p q , so we get a contradiction. If a R H v p x q X H v p x q , the firstpoint on the segment from a to a ´ C v { C that is not in D is in Λ and in Q p C q , hence acontradiction. Claim 3.
The set r K s B zr G s B is within distance C of D . Proof of Claim 3.
By Claim 2 we have K “ D Y G Ą K . It then clearly suffices to provethat r K s B zr G s B is within distance C of D .Consider a crumb G P G at distance at most C from D , so at distance at most C from a shrunken droplet D and necessarily at distance at least C { C ´ C ´ C from anyother shrunken droplet and from B if D is a DYD. By Lemma 5.16 r G s D “ r G s D YB is withindistance C of G . Hence, r K Y Bs “ D Y B Y r G s Y ď G,D r G s D , (9)where the last union is on couples p G, D q as above. Indeed, all r G s D and r G s (for different G )are at distance at least C ´ C from each other and from D z D (by the reasoning above),so for each site of Λ the intersection of the ball of radius O p q centered at it with the set onthe right-hand side of (9) coincides with the intersection with one of the sets r G Y D s , r G s or D Y B , which are all stable, so no infections occur, which proves (9).The claim follows easily from (9), since for every couple
G, D the set r G s D is withindistance C of G , which is itself at distance at most C from D , and G has diameter muchsmaller than C by Corollary 5.14.We thus have that any modified cluster of r K s B is of diameter at most C (by Defini-tion 5.1) and intersects r K s B zr G s B (by Claim 1), which is within distance C of D (byClaim 3). Hence, any such set is within distance C of D .Therefore, Ť C P C pr K s B q Q p C q Ă D Y B , where the union is over all modified clustersof r K s B , since diam p Q p C qq ! C { C ď d p D , Λ z D q . As D is the output of the dropletalgorithm, D is the union of disjoint DYD non-intersecting B and CDYD, so it necessarilycontains Ť D P D D (see Remark 5.7), which concludes the proof. Remark 5.18.
It should be noted that the algorithm is more easily and naturally definedwith no boundary, but that will not be sufficient for our purposes. However, this ‘free’algorithm is trivially obtained as a specialisation of ours. It is also possible to deal withmore general boundaries, with infinite input sets, as well as with droplets defined by moredirections and possibly with several rugged sides.
In this section we map the original dynamics into an East one and conclude the proof of ourmain result. In Section 6.1 we introduce the necessary notation for the relevant geometry. InSection 6.2 we consider a renormalised dynamics on the slices of Figure 6 by algorithmicallyselecting certain modified spanned droplets of size Ω p { q α q . In Section 6.3 we further renor-malise to recover an exact East dynamics where q is replaced by q eff corresponding to theprobability of spanning such a droplet. Finally, in Section 6.4 we prove Theorem 2.8 roughlyas in [25]. 22 u v v u u u C Figure 6: The domain V isthe thickened triangle, a por-tion of which is displayed.Solid lines separate columns C i . Inside the domain isdrawn a DYD, which wit-nesses Φ p ω q “Ò . Set L “ {p C q α q and ι “ min t x ě x {p q α q u P Z u , so that ι “ ` O p q α q . We consider atriangular domain V (see Figure 6), V “ H u p e L u qz ` H u p´ ι {p q α q u q Y H u p´ ι {p q α q u q ˘ . Let us choose C so that N “ e L q α {p ι q ` { “ e L q α p { ` O p q α qq is an integer. We then partition the domain V “ Ť Ni “ C i into regions with C i “ t x P V : e L ´ ι p i ´ q{ q α ą x x, u y ě e L ´ ιi { q α u , so that is in the middle of C N and e L u P Z . We shall refer to C i as the i -th column .Finally, set H i “ H u pp e L ´ ιi { q α q u q and B i “ H i Y ¯ B where ¯ B “ H u p´ ι {p q α q u q Y H u p´ ι {p q α q u q . Note that these boundaries are of the form considered in Section 5.
Let ω P Ω . We will now define a collection of arrow variables which depend only on therestriction of ω to V . We naturally identify the restriction of ω to V with the subset of V where ω is and we use the notation ω “ ∅ to indicate that all sites are filled (healthy) in V , namely ω x “ for all x P V . Let ω p q “ ω X V . We call position of the first up-arrow thesmallest index i p ω q P t , , . . . , N u such that there is a modified spanned droplet of sizeat least L for r ω p q s B i p ω q with boundary B i p ω q . If no such i exists, we say that there are noup-arrows and set i p ω q “ 8 . We further denote ω p q “ ω p q X H i p ω q as soon as i p ω q ă 8 ,while otherwise ω p q “ ∅ .We define the set I p ω q “ t i p ω q , i p ω q , . . . u Ă t , . . . , N u containing the positions ofup-arrows recursively as follows. If there are no up-arrows, then I “ ∅ . Otherwise, we set I p ω q “ t i p ω qu Y I p ω p q q and ω p k q “ p ω p k ´ q q p q , which defines ω p k q for all k . Let us note that23f i p ω q ‰ 8 , then i p ω q ă i p ω p q q , since by definition r ω p q s B i p ω q “ ∅ . Finally, we define Φ p ω q P tÒ , Óu t ,..., N u as Φ p ω q k “ Ò if k P I p ω q , Ó otherwise.The next Lemma states that the probability to find at least one up-arrow decays as q eff “ e ´ L . Lemma 6.1. µ p i ă 8q ď q eff . Proof.
Fix ď i ď N and consider the event i “ i . It is clearly included in the event E i that there is a modified spanned droplet of size at least L for r ω p q s B i with boundary B i .By Proposition 5.17 there is also a spanned droplet of size at least L { C for ω p q zB i withboundary B i . By Lemma 5.10 this implies that there is also a spanned connected droplet ofsize between L { C and L { C . Then one can rewrite E i as the union over all such droplets D of the event that D is spanned. Note that for each discretised DYD D X Z the eventthat there exists a spanned DYD D with D X Z “ D X Z coincides with the event that asuitably chosen such D is spanned. Indeed, the intersection of two DYD is a DYD by (7) andthe spanning of all D depend only on the finite number of sites in D X Z , so there is a finitenumber of possible events associated to different D and one can consider the intersectionof a D defining each of these events. The same reasoning holds for CDYD and so for eachdiscretised droplet D X Z one can bound the probability that there exists a spanned dropletwith such discretisation using Lemma 5.12. Thus, by the union bound on discretised dropletscounted in Observation 5.5, one obtains µ p E i q ď | V | .e L e ´ C L { C ď q eff {p N q . We next consider the event of having at least n up-arrows B p n q “ t ω P Ω : | I p ω q| ě n u . Corollary 6.2.
For any ď n ď N we have µ p B p n qq ď q n eff . Proof.
We prove the statement by induction on n . The base, n “ , is given by Lemma 6.1.For n ą we have µ p| I | ě n q “ N ÿ i “ µ p i p ω q “ i ; | I p ω X H i q| ě n ´ qď N ÿ i “ µ p i “ i q µ p| I | ě n ´ qď q eff n , where we used that the event i “ i only depends on ω z H i ( i is a stopping time for thefiltration induced by the columns) and that the event | I | ě n ´ is increasing for the orderdefined by ω ĺ ω when ω Ă ω . 24e will now state a key deterministic property of the arrows under legal moves of theKCM dynamics. Lemma 6.3.
Let ω P Ω . Let x P C i be such that ω x “ and the constraint at x is satisfiedby ω Y ¯ B . Assume that Φ p ω q ‰ Φ p ω x q . Let j “ max t k : Φ p ω q k ‰ Φ p ω x q k u . Then Φ p ω q r i ´ ,j s “ pÒ , Ó , Ò , Ó , Ò , . . . q , Φ p ω x q r i ´ ,j s “ pÒ , Ò , Ó , Ò , Ó , . . . q and Φ p ω q r ,i ´ s “ Φ p ω x q r ,i ´ s with the convention that Φ p ω q “Ò for all ω .Proof. We denote
Φ : “ Φ p ω q and Φ : “ Φ p ω x q . Clearly, Φ r ,i ´ s “ Φ ,i ´ s , since those valuesdo not depend on ω X H i ´ . Claim 1.
Let k ě i . If Φ k “Ò , then Φ r k ` , N s ě Φ k ` , N s for the lexicographic orderassociated to ÒăÓ . If Φ k “Ò , then Φ r k ` , N s ď Φ k ` , N s . Proof of Claim 1.
The two assertions being analogous, we only prove the first one, so assumethat Φ k “Ò . Let j “ min t l ą k : Φ l “Òu . Then there is a modified spanned droplet of sizeat least L for r ω p q X H k s B j with boundary B j . But this is also true for ω x instead of ω , asthey coincide in H k , and in particular the position of the first up-arrow of Φ after k is atmost j . Claim 2.
Let k ě i ´ be such that Φ k “ Φ k “Ó . Then k ą j i.e. Φ r k, N s “ Φ k, N s . Proof of Claim 2.
We can clearly assume that k ă N . Further assume for a contradictionthat Φ k ` “Ò and Φ k ` “Ó . Let i “ max t l ă k : Φ l “Òu . Then there exists a modifiedspanned droplet D of size at least L for r ω p q X H i s B k ` with boundary B k ` . By Lemma 5.10we can assume that L ď | D | ď C L . However, if d p D, C k ` q ą C , then D is also modifiedspanned for r ω p q X H i s B k with boundary B k , contradicting the definition of i . Indeed, fromthe output of the modified droplet algorithm for r ω p q X H i s B k X D with boundary B k we cancreate a collection ˆ D of droplets for B k ` by extending CDYD appropriately, thus ˆ D contains Q p C qzB k “ Q p C qzB k ` for every modified cluster C of r ω p q X H i s B k X D . Moreover, themodified clusters of r ω p q X H i s B k ` X D are contained in the modified clusters of r ω p q X H i s B k X D , so ˆ D contains the output of the modified droplet algorithm for r ω p q X H i s B k ` X D with boundary B k ` by Remark 5.7, itself containing D .Therefore, d p D, C k ` q ď C . Moreover, D is not modified spanned for rp ω x q p q X H k ´ s B k ` with boundary B k ` (otherwise Φ k,k ` s ‰ pÓ , Óq ). Therefore, there exists a site y P D suchthat y P r ω p q X H i s B k ` zrp ω x q p q X H k ´ s B k ` . We consider two subcases. First assume that d p x, R z H i ´ q ě C . Then, the constraintat x is satisfied by p ω X H i ´ q Y ¯ B , so r ω p q X H k ´ s B k ` “ rp ω x q p q X H k ´ s B k ` , and there is apath P Ă r ω p q X H i s B k ` zrp ω x q p q X H k ´ s B k ` from R z H k ´ to y such that each two consecutive sites are at distance at most O p q . But d p y, R z H k ´ q ě ι { q α ´ diam p D q ´ C ě C p L ` q , so one can find a subpath P Ă C k X P of diameter at least C L . Yet, it is clear that P Ă r ω p q X H i s B k implies the existence ofa modified spanned droplet of size larger than L with boundary B k , so one would have anup-arrow of Φ in r i ` , k s – a contradiction. If, on the contrary, d p x, R z H i ´ q ď C , we canredo the same reasoning, but P needs to extend to either R z H k ´ or x , both of which aresufficiently far from y . 25hus, Φ k ` “ Φ k ` , as the case Φ k ` “Ó , Φ k ` “Ò is treated identically. But theneither both are Ò , in which case we are done by Claim 1 or both are Ó and we are done byinduction.It is easy to see that the only non-identical arrow sequences Φ r i ´ ,j s and Φ i ´ ,j s satisfyingthe two claims are pÒ , Ó , Ò , Ó , . . . q and pÒ , Ò , Ó , Ò , . . . q (in this order using that ω x “ ). Indeed,by Claims 1 and 2 Φ k ‰ Φ k for all i ď k ď j , by Claim 1 one cannot have two consecutiveup arrows neither in Φ nor in Φ in the interval r i, j s and by Claim 2 Φ i ´ “ Φ i ´ “Ò . We partition t , . . . , N u into blocks B i “ t i ´ , i u for ď i ď N . Given ω P Ω , we define η p ω q P t , u t ,...,N u by η p ω q i “ t@ j P B i :Φ p ω q j “Óu for all i P t , . . . N u . Let n “ t L u “ Z C q α ^ ă t log N u . Recall the definition of legal paths, Definition 2.4. Given an event E Ă Ω and a legalpath γ “ p ω p q , . . . , ω p k q q we will say that γ X E “ ∅ if ω p i q R E for all i P t , . . . , k u . Also,given ω P Ω and A Ă Ω , we say that γ connects ω to A if ω p q “ ω and ω p k q P A . Recallthat B p n q Ă Ω is the set of configurations with at least n up-arrows. The following is astraightforward but important corollary of Lemma 6.3. Corollary 6.4.
For any legal path p ω p q , . . . , ω p k q q , the path p η p ω p q q , . . . , η p ω p k q qq is legal forthe East model on t , . . . , N u defined by fixing η “ .Proof. By Lemma 6.3 η p ω p j q q ‰ η p ω p j ` q q implies that Φ p ω p j q q and Φ p ω p j ` q q only differ onan alternating chain of arrows ending in some B i , preceded by Ò . Then clearly η p ω p j q q l “ η p ω p j ` q q l for all l ‰ i and η p ω p j q q i ´ “ .Let Ω Ó and Ω N Ò be respectively the set of configurations which do not have up-arrows,and the set of configurations with an up-arrow in the N -th column, namely Ω Ó “ t ω P Ω : Φ p ω q “ pÓ , . . . , Óqu , Ω N Ò “ t ω P Ω : Φ p ω q N “Òu . Combining the last corollary with Proposition 2.7, we obtain the most important inputfor the proof of the main result.
Corollary 6.5.
For any ω P Ω Ó there does not exist a legal path γ with γ X B p n ` q “ ∅ connecting ω to Ω N Ò . To prove Theorem 2.8 it is sufficient to prove the lower bound for the mean infection timeand use the following inequality (see [10, Theorem 4.4] and also [26, Section 2.2]) T rel ě q E p τ q . (10)However, it is instructive to construct at this stage a test function that directly gives thedesired lower bound on T rel without going through the comparison with the mean infectiontime. Indeed, the mechanism will appear more clearly this way.26 roof of Theorem 2.8 for T rel We define the event ˜ A “ t ω P Ω : D a legal path γ with γ X B p n q “ ∅ connecting ω Y p Z z V q to Ω Ó u and the test function f : Ω Ñ t , u f “ ˜ A . Then, by Definition 2.5 we get T rel ě µ p ˜ A qp ´ µ p ˜ A qq D p f q , (11)where the Dirichlet form D p f q is defined in (4). Lemma 6.6 (Bounds on µ p ˜ A q ) . µ p ˜ A q ´ ´ µ p ˜ A q ¯ ě exp ˆ log qC q α ˙ . Proof.
By Lemma 6.1 we have µ p ˜ A q ě µ p Ω Ó q ě ´ q eff ě { . On the other hand, ´ µ p ˜ A q ě µ p Ω N Ò q ě q C L ě exp p C log q {p C q α qq , where we used Corollary 6.5 for the first inequality as well as the fact that if p ω p q , . . . , ω p k q q is a legal path, then p ω p k q , . . . , ω p q q is one as well, and for the second inequality we noticethat for the N -th arrow to be up it is sufficient to have an empty segment of length C L in C N . Lemma 6.7 (Estimate of the Dirichlet form) . D p f q ď exp p´ {p C q α qq .Proof. Using the fact that f p ω q depends only on the values of ω in V , we get D p f q “ ÿ x P V µ p c x Var x p f qq “ q p ´ q q ÿ x P V µ ´ c x t ω P ˜ A , ω x R ˜ A u ` c x t ω R ˜ A , ω x P ˜ A u ¯ ď | V | µ p B p n ´ qq , (12)since, by Lemma 6.3 || I p ω q| ´ | I p ω x q|| ď when c x “ , so the indicators both imply ω P B p n ´ q . Indeed, ω P ˜ A implies the existence of a legal path γ from Ω Ó to ω Y p Z z V q with each configuration not in B p n q . Since c x “ , the path ¯ γ obtained by adding thetransition from ω Y p Z z V q to ω x Y p Z z V q is also legal, thus the hypothesis ω x R ˜ A is notsatisfied unless ω x P B p n q (and similarly for ω R ˜ A , ω x P ˜ A ). Thus, the result follows by usingCorollary 6.2.Then the lower bound for T rel of Theorem 2.8 follows from (11), Lemma 6.6 and Lemma6.7.The above proof, together with the matching upper bound of Theorem 2(a) of [26] indicatethat the bottleneck dominating the time scales is the creation of Θ p log p { q eff qq simultaneousdroplets of probability q eff . 27 roof of Theorem 2.8 for E p τ q The proof of the lower bound for the infection timefollows a similar route, with some complications due to the fact that we have to identifya (sufficiently likely) initial set starting from which we have to go through the bottleneckconfigurations before infecting the origin.By [25, Corollary 3.4], to prove the desired lower bound on E p τ q it suffices to constructa local function φ “ φ q such that(i) µ p φ q “ ,(ii) µ p φ q D p φ q ě exp p {p C q α qq ,(iii) φ p ω q “ if ω “ .Inspired by [25] we let Ω g “ Ω Ó X t ω P Ω : ω Λ “ u where Λ “ t x P Z : d p x, q ď {p q α qu Ă C N and A “ t ω P Ω : D a legal path γ with γ X B p n q “ ∅ connecting ω Y p Z z V q to Ω g u . Then we set φ p¨q “ A p¨q{ µ p A q { . (13)We are now left with proving that this function satisfies (i)-(iii) above.Property (i) follows immediately from (13). In order to verify (ii) we start by establishinga lower bound on µ p A q . By definition it holds that µ p A q ě µ p Ω g q ě µ p ω Λ “ q µ p Ω Ó q ě e ´ O p q{ q α ´ p ´ q eff q “ e ´ O p q{ q α ´ , (14)where we used Harris’ inequality [18] ( t ω Λ “ u and Ω Ó are increasing events if we considerthat ω ď ω when ω x ď ω x for all x P Z ), Lemma 6.1 and | Λ | “ O p { q α q .Furthermore, one can repeat the proof of Lemma 6.7 to obtain D p φ q ď e ´ {p C q α q . (15)Thus, recalling (14), Property (ii) holds.We are therefore only left with proving the next lemma establishing Property (iii), com-pleting the proof of Theorem 2.8. Lemma 6.8.
Let ω be such that ω “ . Then any legal path connecting Ω g to ω intersects B p n q . As in the lower bound on ´ µ p ˜ A q for T rel , the proof relies on Corollary 6.5, but anadditional complication arises due to the fact that emptying the origin does not a priorirequire creating a critical droplet nearby. Proof of Lemma 6.8.
Suppose for a contradiction that there exists a configuration ω with ω “ , a configuration ω p q P Ω g and a legal path γ “ p ω p q , . . . , ω p k q q with ω p k q “ ω and ω p j q R B p n q for all j P t , . . . , k u . Assuming without loss of generality that ω p j q ‰ ω p j ´ q forall j , let x j be such that ω p j q “ p ω p j ´ q q x j . Consider the path ˜ γ “ p ˜ ω p q , . . . , ˜ ω p k q q obtained byperforming the same updates as for γ except for flips in the column C N , which are performedonly if they correspond to emptying sites. More precisely, we let ˜ ω p q “ ω p q and ˜ ω p j q “ p ˜ ω p j ´ q q x j if x j R C N or p ˜ ω p j ´ q q x j “ , ˜ ω p j ´ q otherwise.28t is not difficult to verify by induction that ˜ γ is also a legal path with ˜ ω p j q ď ω p j q for all j (where ω ď ω when ω x ď ω x for all x P Z ) and that ˜ ω p j q and ω p j q coincide outside of C N .Then p ˜ ω p k q q ď p ω p k q q “ and by definition p ˜ ω p q q Λ “ . Therefore, since inside C N eachsite that has been emptied in γ is also empty in ˜ ω p k q , we conclude that necessarily ˜ ω p k q X C N contains a (modified) spanned droplet of size {p C q α q ą L with boundary B N “ ¯ B . Indeed,there is a path of sites x with steps of size O p q from Z z Λ to such that p ˜ ω p k q q x “ . Thismeans that ˜ ω p k q P Ω N Ò . Furthermore, for all j we have Φ p ˜ ω p j q q r , N ´ s “ Φ p ω p j q q r , N ´ s , asthose do not depend on the sites in C N . Thus, using Corollary 6.5, together with the factsthat ˜ ω p q P Ω g Ă Ω Ó , ˜ ω p k q P Ω N Ò and ˜ γ X B p n ` q “ ∅ , we reach a contradiction. With Theorem 2.8 the scaling of the infection time is determined up to a polylogarithmicfactor. The next natural question is to pursue determining this factor in the spirit of therefined universality result of [7]. For the moment there is only one critical model withinfinitely many stable directions for which this is known — the Duarte model [25]. In thatcase the corrective factor is Θ pp log q q q . However, for bootstrap percolation there are alreadytwo different possible behaviours of this factor depending on whether the model is balancedor unbalanced (see Definition 2.3). Based on this one could expect the following. Conjecture 7.1.
Let U be a critical update family with an infinite number of stable directions. • If U is balanced, then E p τ q “ exp ˆ Θ p q q α ˙ . • If U is unbalanced, then E p τ q “ exp ˜ Θ ` p log q q ˘ q α ¸ . The same asymptotics hold for T rel . In other words we expect the lower bound of Theorem 2.8 to be sharp for balanced models,while the upper bound of [26, Theorem 2(a)] to be sharp for unbalanced ones. The balancedcase is not hard and only requires an improvement of the approach of [26]. It will be treatedin a future work, since it shares none of the techniques discussed here. In the unbalanced casethe p log q q should arise as the square of the p log q q factor for bootstrap percolation, itselfcaused by the one-dimensional geometry and larger size of critical droplets. This is indeedwhat happens for the Duarte model [25], an example of unbalanced critical constraint. Acknowledgements
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