Universality for critical kinetically constrained models:
infinite number of stable directions
Kinetically constrained models (KCM) are reversible interacting particle systems on with continuous-time constrained Glauber dynamics. They are a natural non-monotone stochastic version of the family of cellular automata with random initial state known as -bootstrap percolation. KCM have an interest in their own right, owing to their use for modelling the liquid-glass transition in condensed matter physics.
In two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes. Here we study in full generality the class termed ‘critical’. Together with the companion paper by Martinelli and two of the authors  we establish the universality classes of critical KCM and determine within each class the critical exponent of the infection time as well as of the spectral gap. In this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart. This is due to the occurrence of ‘energy barriers’, which determine the dominant behaviour for these KCM but which do not matter for the monotone bootstrap dynamics. Our result confirms the conjecture of Martinelli, Morris and the last author , who proved a matching upper bound.
MSC2010: Primary 60K35; Secondary 82C22, 60J27, 60C05
Keywords: Kinetically constrained models, bootstrap percolation, universality, Glauber dynamics, spectral gap.
Kinetically constrained models (KCM) are interacting particle systems on the integer lattice , which were introduced in the physics literature in the 1980s by Fredrickson and Andersen  in order to model the liquid-glass transition (see e.g. [31, 17] for reviews), a major and still largely open problem in condensed matter physics . A generic KCM is a continuous-time Markov process of Glauber type characterised by a finite collection of finite nonempty subsets of , its update family. A configuration is defined by assigning to each site an occupation variable corresponding to an empty or occupied site respectively. Each site waits an independent, mean one, exponential time and then, iff there exists such that for all , site is updated to empty with probability and to occupied with probability . Since each is contained in , 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() measure, , which is therefore a reversible invariant measure. Hence the process started at is stationary.
Both from a physical and from a mathematical point of view, a central issue for KCM is to determine the speed of divergence of the characteristic time scales when . Two key quantities are: (i) the relaxation time , i.e. the inverse of the spectral gap of the Markov generator (see Definition 2.5) and (ii) the mean infection time , i.e. the mean over the stationary process started at 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 choices of the constraints [2, 9, 25, 12, 13, 27] (see also  section 1.4.1 for a non exhaustive list of references in the physics literature). These results show that KCM exhibit a very large variety of possible scalings depending on the update family . A question that naturally emerges, and that has been first addressed in , is whether it is possible to group all possible update families into distinct universality classes so that all models of the same class display the same divergence of the time scales.
Before presenting the results and the conjectures of , we should describe the key connection of KCM with a class of discrete monotone cellular automata known as -bootstrap percolation (or simply bootstrap percolation) . For -bootstrap percolation on , given an update family and a set of sites infected at time , the infected sites in remain infected at time , and every site becomes infected at time if the translate by of one of the sets in is contained in . The set of initial infections is chosen at random with respect to the product Bernoulli measure with parameter , which identifies with : for every we have . One then defines the critical probability to be the infimum of the such that with probability one the whole lattice is eventually infected, namely . A key time scale for this dynamics is the first time at which the origin is infected, . In order to study this infection time for models on , the update families were classified by Bollobás, Smith and Uzzell  into three universality classes: supercritical, critical and subcritical, according to a simple geometric criterion (see Definition 2.1). In  they proved that if is supercritical or critical, and it was proved by Balister, Bollobás, Przykucki and Smith  that if is subcritical. For supercritical update families,  proved that w.h.p. as , while in the critical case . The result for critical families was later improved by Bollobás, Duminil-Copin, Morris and Smith , who identified the critical exponent such that .
Back to KCM, if we fix an update family and an initial configuration and we identify the empty sites with infected sites, a first basic observation is that the clusters of sites that will never be infected in the -bootstrap percolation correspond to clusters of sites which are occupied and will never be emptied under the KCM dynamics. A natural issue is whether there is a direct connection between the infection mechanism of bootstrap percolation and the relaxation mechanism for KCM, and, more precisely, whether the scaling of and is connected to the typical value of when the law of the initial infections is . It is not difficult to establish that provides a lower bound for and (see [27, Lemma 4.3] and (10)), but in general, as we will explain, this lower bound does not provide the correct behaviour.
In , Martinelli, Morris and the last author proposed that the supercritical class should be refined into unrooted supercritical and rooted supercritical models in order to capture the richer behavior of KCM. For unrooted models the scaling is of the same type as for bootstrap percolation, as [26, Theorem 1(a)]111For the lower bound of one does not need to use the boostrap percolation results, as by plugging the test function in Definition 2.5., while for rooted models the divergence is much faster, (see [26, Theorem 1(b)] for the upper bound and [25, Theorem 4.2] for the lower bound).
Concerning the critical class, the lower bound with mentioned above and the results of  on bootstrap percolation imply that and diverge at least as . 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 such that both and scale as , with in general different from the exponent of the corresponding bootstrap percolation process. Furthermore, a toolbox was developed for the study of the upper bounds, leading to upper bounds matching this conjecture for all models. The main issue left open in  was to develop tools to establish sharp lower bounds. A first step in this direction was done by Martinelli and the last two authors  by analyzing a specific critical model known as the Duarte model for which the update family contains all the -elements subsets of the North, South and West neighbours of the origin. Theorem 5.1 of  establishes a sharp lower bound on the infection and relaxation times for the Duarte KCM that, together with the upper bound in [26, Theorem 2(a)], proves as , and the same result holds for . The divergence is again much faster than for the corresponding bootstrap percolation model, for which it holds w.h.p as  (see also , from which the sharp value of the constant follows), namely the critical exponent for the Duarte KCM is twice the critical exponent for the Duarte bootstrap percolation.
Both for Duarte and for supercritical rooted models, the sharper divergence of time scales for KCM is due to the fact that the infection time of KCM is not well approximated by the infection mechanism of the monotone bootstrap percolation process, but is instead the result of a much more complex infection/healing mechanism. Indeed, visiting regions of the configuration space with an anomalous amount of empty sites is heavily penalised and requires a very long time to actually take place. The basic underlying idea is that the dominant relaxation 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 adjacent droplet but only in a certain direction (or more precisely in a limited cone of directions). This is reminiscent of the relaxation mechanism for the East model, a prototype one-dimensional KCM for which can be updated iff is empty, thus a single empty site allows to create/destroy an empty site only on its right (see  for a review on the East model). For supercritical rooted models, the empty droplets that play the role of the single empty sites for East have a finite (model dependent) size, hence an equilibrium density . For the Duarte model, droplets have a size that diverges as and thus an equilibrium density . Then a (very) rough understanding of the results of [25, 26] is obtained by replacing with in the time scale for the East model . The main technical difficulty to translate this intuition into a lower bound is that the droplets cannot be identified with a rigid structure. In  this difficulty for the Duarte model was overcome by an algorithmic construction that allows to sequentially scan the system in search of sets of empty sites that could (without violating the constraint) empty a certain rigid structure. These are the droplets that play the role of the empty sites for the East dynamics.
In  all critical models which have an infinite number of stable directions (see Section 2.1), of which the Duarte model is but one example, were conjectured to have a critical exponent , with the critical exponent of the corresponding bootstrap percolation dynamics (defined in Definition 2.2). The heuristics is the same as for the Duarte model, the only difference being that droplets would have in general size . However, the technique developed in  for the Duarte model relies heavily on the specific form of the Duarte constraint and in particular on its oriented nature222Note that, since the Duarte update rules contain only the North, South and West neighbours of the origin, the constraint at a site does not depend on the sites with abscissa larger than the abscissa of ., and it cannot be extended readily to this larger class.
In this work, together with the companion paper by Martinelli and two of the authors , we establish in full generality the universality classes for critical KCM, determining the critical exponent for each class.
Here we treat all choices of for which there is an infinite number of stable directions and prove (Theorem 2.8) a lower bound for and that, together with the matching upper bound of [26, Theorem 2], yields
for and the same result for . Our technique is somewhat inspired by the algorithmic construction of , however, the nature of the droplets which move in an East-like way is here much more subtle, and in order to identify them we construct an algorithm which can be seen as a significant improvement on the -covering and -iceberg algorithms developed in the context of bootstrap percolation .
In the companion paper  we prove for the complementary class of models, namely all critical models with a finite number of stable directions, an upper bound that (together with the lower bound from bootstrap percolation) yields instead
for and the same result for .
A comparison of our results with Conjecture 3 of  is due. The class that we consider here is, in the notation of , the class of models with bilateral difficulty , hence belong to the -rooted class defined therein. Therefore, our Theorem 2.8 proves Conjecture 3(a) in this case. We underline that it is not a limitation of our lower bound strategy that prevents us from proving Conjecture 3(a) for the other -rooted models, namely those with . Indeed, as it is proven in the companion paper , in this case the conjecture of  is not correct, since it did not take into account a subtle relaxation mechanism which allows to recover the same critical exponent as for the bootstrap percolation dynamics.
The plan of the paper is as follows. In Section 2 we develop the background for both KCM and bootstrap percolation needed to state our result, Theorem 2.8. In Section 3 we give a sketch of our reasoning and highlight the important points. In Section 4 we gather some preliminaries and notation. Section 5 is the core of the paper — there we define the central notions and establish their key properties, culminating in the Closure Proposition 5.17. In Section 6 we establish a connection between the KCM dynamics and an East dynamics and use this to wrap up the proof of Theorem 2.8. Finally, in Section 7 we discuss some open problems.
2 Models and background
2.1 Bootstrap percolation
Before turning to our models of interest, KCM, let us recall recent universality results for the intimately connected bootstrap percolation models in two dimensions. -bootstrap percolation (or simply bootstrap percolation) is a very general class of monotone transitive local cellular automata on first studied in full generality by Bollobás, Smith and Uzzell . Let , called update family, be a finite family of finite nonempty subsets, called update rules, of . Let , called the set of initial infections, be an arbitrary subset of . Then the -bootstrap percolation dynamics is the discrete time deterministic growth of infection defined by and, for each ,
In other words, at any step each site becomes infected if a rule translated at it is already fully infected, and infections never heal. We define the closure of the set by and we say that is stable when . The set of initial infections is chosen at random with respect to the product Bernoulli measure with parameter : for every we have .
Arguably, the most natural quantity to consider for these models is the typical (e.g. mean) value of , the infection time of the origin.
The combined results of Bollobás, Smith and Uzzell  and Balister, Bollobás, Przykucki and Smith  yield a pre-universality partition of all update families into three classes with qualitatively different scalings of the median of the infection time as . In order to define this partition we will need a few definitions.
For any unitary vector ( denotes the Euclidean norm in ) and any vector we denote — the open half-plane directed by passing through . We also set . We say that a direction is unstable (for an update family ) if there exists such that and stable otherwise. The partition is then as follows.
Definition 2.1 (Definition 1.3 of ).
An update family 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 number of stable directions,
The main result of  then states that in the supercritical case with high probability as , while in the critical one . The final justification of the partition in Definition 2.1 was given by Balister, Bollobás, Przykucki and Smith  who proved that the origin is never infected with positive probability for subcritical models for sufficiently small, i.e. if is subcritical. From the bootstrap percolation perspective supercritical models are rather simple, while subcritical ones remain very poorly understood (see ). Nevertheless, most of the non-trivial models considered before the introduction of -bootstrap percolation, including the -neighbour model (see [1, 22] for further results), fall into the critical class, which is also the focus of our work.
Significantly improving the result of , Bollobás, Duminil-Copin, Morris and Smith  found the correct exponent determining the scaling of for critical families. Moreover, they were able to find up to a constant factor. To state their results we need the following crucial notion.
Definition 2.2 (Definition 1.2 of ).
Let be an update family and be a direction. Then the difficulty of , , is defined as follows.
If is unstable, then .
If is an isolated stable direction (isolated in the topological sense), then
i.e. the minimal number of infections allowing to grow infinitely.
We define the difficulty of by
where is the set of open semi-circles of .
It is not hard to see (Theorem 1.10 of , Lemma 2.6 of ) that the set of stable directions is a finite union of closed intervals of and that (Lemmas 2.7 and 2.10 of ) (1) also holds for unstable and strongly stable directions, that is directions in the interior of the set of stable directions (but not for semi-isolated stable directions i.e. endpoints of non-trivial stable intervals). Furthermore (see [7, Lemma 2.7], [8, Lemma 5.2]), if and only if is an isolated stable direction, so that is critical if and only if . As a final remark we recall that, contrary to determining whether an update family is critical, finding is a NP-hard question .
We are now ready to describe the universality results. A weaker form of the result of  is that with high probability as . For the full result however, we need one last definition.
A critical update family is balanced if there exists a closed semi-circle such that and unbalanced otherwise.
2.2 Kinetically constrained models
Returning to KCM, let us first define the general class of KCM introduced by Cancrini, Martinelli, Roberto and the last author  directly on . Fix a parameter and an update family as in the previous section. The corresponding KCM is a continuous-time Markov process on which can be informally defined as follows. A configuration is defined by assigning to each site an occupation variable corresponding to an empty (or infected) and occupied (or healthy) site respectively. Each site waits an independent exponentially distributed time with mean before attempting to update its occupation variable. At that time, if the configuration is completely empty on at least one update rule translated at , i.e. if such that for all , then we perform a legal update or legal spin flip by setting to with probability and to with probability . Otherwise the update is discarded. Since the constraint to allow the update never depends on the state of the to-be-updated site, the product measure is a reversible invariant measure and the process started at is stationary. More formally, the KCM is the Markov process on with generator acting on local functions as
where denotes the average of with respect to the variable conditionally on , and is the indicator function of the event that there exists such that is completely empty, i.e. . We refer the reader to chapter I of , where the general theory of interacting particle systems is detailed, for a precise construction of the Markov process and the proof that is the generator of a reversible Markov process on with reversible measure .
The corresponding Dirichlet form is defined as
where denotes the variance of the local function with respect to the variable conditionally on . The expectation with respect to the stationary process with initial distribution will be denoted by . Finally, given a configuration and a site , we will denote by the configuration obtained from by flipping site , namely by setting and for all . For future use we also need the following definition of legal paths, that are essentially sequences of configurations obtained by successive legal updates.
Definition 2.4 (Legal path).
Fix an update family , then a legal path in is a finite sequence such that, for each the configurations and differ by a legal (with respect to the choice of ) spin flip at some vertex .
As mentioned in Section 1, our goal is to prove sharp bounds on the characteristic time scales of critical KCM. Let us start by defining precisely these time scales, namely the relaxation time (or inverse of the spectral gap) and the mean infection time (with respect to the stationary process).
Definition 2.5 (Relaxation time ).
Given an update family and , we say that is a Poincaré constant for the corresponding KCM if, for all local functions , we have
If there exists a finite Poincaré constant, we define
Otherwise we say that the relaxation time is infinite.
A finite relaxation time implies that the reversible measure is mixing for the semigroup with exponentially decaying time auto-correlations (see e.g. [3, Section 2.1]).
Definition 2.6 (Infection time ).
The random time at which the origin is first infected is given by
where we adopt the usual notation letting be the value of the configuration at the origin, namely .
The East model
We close this section by defining a specific example of KCM on , the East model of Jäckle and Eisinger , which will be crucial to understand our results (KCM on are defined in the same way as KCM on ). It is defined by an update family composed by a single rule containing only the site to the left of the origin (). In other words, site can be updated iff is empty. For this model both and scale as as [2, 9, 12]333Actually these references focus on the study of . A matching upper bound for follows from (10). The lower bound for follows easily from the lower bound for with obtained in the proof of Theorem 5.1 of .. One of the key ingredients behind this scaling is the following combinatorial result  (see [14, Fact 1] for a more mathematical formulation).
Consider the East model on defined by fixing at all time. Then any legal path connecting the fully occupied configuration (namely s.t. for all ) to a configuration such that goes through a configuration with at least empty sites.
This logarithmic ‘energy barrier’, to employ the physics jargon, and the fact that at equilibrium the typical distance to the first empty site is are responsible for the divergence of the time scales as roughly .
In this paper we study critical KCM with an infinite number of stable directions or, equivalently, with a non-trivial interval of stable directions.
Let be a critical update family with an infinite number of stable directions. Then there exists a sufficiently large constant such that
as and the same asymptotics holds for .
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 in the sense of Corollary 2.9 below. We thus complete the proof of universality and Conjecture 3(a) of  for these models444The conjecture involuntarily asks for a positive power of , which we do not expect to be systematically present (see Conjecture 7.1)..
Let be a critical update family with an infinite number of stable directions. Then
as and the same holds for .
Universality for the remaining critical models is proved in a companion paper by Martinelli and the first and third authors  and, in particular, Conjecture 3(a) of  is disproved for models other than those covered by Theorem 2.8. It is important to note that Theorem 2.8 significantly improves the best known results for all models with the exception of the recent result of Martinelli and the last two authors  for the Duarte model. Indeed, the previous bound had exponent , and was proved via the general (but in this case far from optimal) lower bound with the mean infection time for the corresponding bootstrap percolation model [27, Lemma 4.3].
3 Sketch of the proof
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 stable directions. We can then construct stable ‘droplets’ of shape as in Figure 3 (see Definitions 5.2 and 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 KCM dynamics. The relevance and advantage of such shapes come from the fact that only infections situated to the left of a droplet can induce growth left. This is manifestly not feasible without the 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 the dynamics. 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 update family with rules ,, (it can also be seen as the modified Duarte model with an additional rule). The stable sets in this case are actually Young 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 of the -covering algorithm of Bollobás, Duminil-Copin, Morris and Smith . It is important for us not to overestimate the closure as brutally. Indeed, a key step and the main difficulty of our work is the Closure Proposition 5.17, which roughly states that the collections of droplets associated to the closure of the initial infections is equal to the collection for the initial infections. This is highly non-trivial, as in order not to overshoot in defining the droplets, one is forced to ignore small patches of infections (larger than the ones in ), which can possibly grow significantly when we take the closure for the bootstrap percolation process and especially so if they are close to a large infected droplet. In order to remedy this problem, we introduce a relatively intrinsic notion of ‘crumb’ (see Definition 5.1) such that its closure remains one and does not differ too much from it. A further advantage of our algorithm for creating the droplets over the one of  is that it is somewhat canonical, with a well-defined unique output, which has particularly nice ‘algebraic’ description and properties (see Remark 5.7). Another notable difficulty we face is systematically working in roughly a 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.
Finally, having established the Closure Proposition 5.17 alongside standard and straightforward results like an Aizenmann-Lebowitz Lemma 5.10 and an exponential decay of the probability of occurrence of large droplets (Lemma 5.12), we finish the proof via the following approach, inspired by the one developed by Martinelli and the last two authors  for the Duarte model. The key step here (see Section 6) is mapping the KCM legal paths to those of an East dynamics via a suitable renormalisation. Roughly speaking, we say that a renormalised site is empty if it contains a large droplet of infections. However, for the renormalised configuration to be mostly invariant under the original KCM dynamics, we rather look for the droplets in the closure of the original set of infections instead. This is where the Closure Proposition 5.17 is used to compensate the fact that the closure of equilibrium is not equilibrium. In turn, this mapping together with the combinatorial result for the East model recalled in Section 2.2 (Proposition 2.7), yield a bottleneck for our dynamics corresponding to the creation of droplets, where is the equilibrium distance between two empty sites in the renormalized lattice, and . This provides for the time scales the desired lower bound of Theorem 2.8. The last part of the proof follows very closely the ideas put forward in  for the Duarte model. However, in , there was no need to develop a subtle droplet algorithm since, owing to the oriented character of the Duarte constraint, droplets could simply be identified with some large infected vertical segments. It is also worth noting that, thanks to the less rigid notion of droplets that we develop in the general setting, some of the difficulties faced in  for Duarte are no longer present here.
4 Preliminaries and notation
Let us fix a critical update family with an infinite number of stable directions for the rest of the paper. We will omit from all notation, such as .
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 and technical, but it simply reflects the fact that we have a lot of freedom for the choice and we make 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 is called rational if .
There exists rational stable directions (see Figure 1) with difficulty at least such that
The directions appear in couterclockwise order .
No is a semi-isolated stable direction.
belongs to the cone spanned by and for i.e. the strictly smaller interval among and contains .
is contained in the interior of the convex envelope of .
Either or .
is stable or, equivalently, .
Since has an infinite number of stable directions and they form a finite union of closed intervals with rational endpoints [8, Theorem 1.10], there exists a non-empty open interval of stable directions. Further note that the set of directions such that there exists a rule and with is finite, so one can find a non-trivial closed subinterval which does not intersect . The directions and will be chosen in , which clearly implies that they are strongly stable and thus with infinite difficulty. Moreover, if there exists with , by stability of , we have , which contradicts .
Since is critical it does not have two opposite strongly stable directions, so there is no strongly stable direction in . If there are any (isolated or semi-isolated) stable directions in , we can further choose a non-trivial open subinterval , for which this is not the case (there is a finite number of isolated and semi-isolated stable directions). Let be such that the angle between any two consecutive directions of difficulty at least is at most (it is well defined by (2)). We then choose a non-trivial closed subinterval with rational and rational and with . It easily follows from the sum and difference formulas for the tangent function that , and are also rational.
These both exist, since does not contain stable directions, both and contain directions with difficulty at least by (2) and the set of such directions is closed. If is not semi-isolated, we set and similarly for . Otherwise, we choose a rational strongly stable direction sufficiently close to as and similarly for . We claim that this choice satisfies all the desired conditions. Indeed, all directions in are stable non-semi-isolated rational with difficulty at least and the last but one condition was already verified.
One does have that is in the cone spanned by and , which is implied by and similarly for , so the third condition is also verified. If , then there is an open half circle contained in with no direction of difficulty at least , which contradicts (2), so and the same holds for , and by the definition of and , the fact that and are sufficiently close to them and the fact that was chosen smaller than . Thus is in the convex envelope of .
Finally, if one has both and , then one obtains , since is smaller than . However, and are consecutive directions of difficulty at least , which contradicts the definition of . ∎
For the rest of the paper we fix directions as in Lemma 4.1 and assume without loss of generality that .
Let us fix large constants
each of which can depend on previous ones as well as on and . We will also use asymptotic notation whose constants can depend on and , but not on or the other constants above. All asymptotic notation is with respect to , so we assume throughout that is sufficiently small.
For any two sets we define .
Finally, we make the convention that throughout the article all distances, balls and diameters are Euclidean unless otherwise stated. We say that a set is within distance of a set if for all where is the Euclidean distance.
5 Droplet algorithm
In this section we define our main tool – the droplet algorithm. It can be seen as a significant improvement on the -covering and -iceberg algorithms [7, Definitions 6.6 and 6.22], many of whose techniques we adapt to our setting.
We will work in an infinite domain defined as follows (see Figure 2). Fix some vector and let
where the directions , and are those defined in Lemma 4.1. In other words, is a cone with sides perpendicular to and cut along a line perpendicular to . The reader is invited to simply think that is a half-plane directed by , which will not change the reasoning.
5.1 Clusters and crumbs
Let be the graph with vertex set but with if and only if . Let be defined similarly with replaced by .
Definition 5.1 (Clusters and crumbs).
Fix a finite set of infected sites. Let be a connected component of the subgraph of induced by . Then is a crumb if it is at distance more than from and there exists a set such that and . Let be a connected component which is not a crumb. We call cluster any such that the induced subgraph of is connected and and such that is maximal with this property. We call boundary cluster every cluster at distance at most from .
We similarly define modified crumb, modified cluster and modified boundary cluster by replacing and by and respectively.
Clearly, any (modified) non-boundary cluster has at least sites. Indeed, if its connected component is of diameter larger than , then the diameter of the cluster is larger than , and we can choose large enough to get , while otherwise the cluster is a connected component which is not a crumb and at distance more than from , so by definition has at least sites. Moreover, a cluster only intersects a bounded number of other clusters, as its diameter is bounded. Also note that crumbs (resp. modified crumbs) are at distance at least (resp. ) from any other site of and have diameter much smaller than , as we shall see in Corollary 5.14. The proofs of this corollary and Observation 5.13 it follows 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 be a cluster (resp. modified cluster). We denote by (resp. ) the smallest open quadrilateral with sides perpendicular to containing the set (resp. ). Note that (resp. ), since (resp. ) is stable and that (resp. ), as . We extend the definition for (non-modified) clusters.
5.2 Distorted Young diagrams
We now define the shape that our ‘droplets’ will have, which resembles Young diagrams555For the 3-rule model alluded to in Section 3 stable sets consist precisely of Young diagrams and the directions provided by Lemma 4.1 can be arbitrarily close to the four axis directions, yielding Young diagrams.. The following definitions are illustrated in Figure 3.
Definition 5.2 (Dyd).
We call distorted Young diagram (DYD) a subset of of the form
for a finite set , some set of vectors and . The vectors and are uniquely defined up to redundancy (and up to the convention that all are on the topological boundary of the DYD). An alternative definition of the DYD can also be given as
where are the convex corners of the diagram rather than the concave ones.
For any DYD we denote by the vector such that
for . We further denote
i.e. the minimal quadrilateral containing with sides directed by . In these terms, (resp. ) is a DYD and .
Definition 5.3 (Cdyd).
We call cut distorted Young diagram (CDYD) a subset of of the form
for a finite set and some vectors and . Alternatively, one can write
where are the convex corners.
For a DYD, , we define as the CDYD defined by the same and or the same . We extend the notation to CDYD by setting if is a CDYD. Note that by Lemma 4.1 all DYD and CDYD are stable for the bootstrap percolation dynamics (restricted to ). Also pay attention to the fact that CDYD are not necessarily connected, contrary to DYD.
Definition 5.4 (Size).
For a DYD we set to be its projection (parallel to ) and to be its size – the length of the projection. For a CDYD we denote its size .
Note that if is a DYD, then by Lemma 4.1 and the assumption we made that . Furthermore, for all DYD again by Lemma 4.1 with constants depending only on . One should be careful with the meaning of size for disconnected CDYD, but it will not cause problems, as all CDYD arising in our forthcoming algorithm are connected.
Note that for any the number of discretised DYD and CDYD (i.e. intersections of a DYD or CDYD with ) containing a fixed point of diameter at most is less than for some constant depending only on .
Note that a DYD or CDYD is uniquely determined by its rugged edge formed by its and -sides. However, this edge injectively defines an oriented percolation path with directions perpendicular to and on the lattice
(except its endpoints, which lie on similar lattices). Since the graph-length of this path is bounded by and its endpoints are within distance from , the result follows. ∎
We next introduce a procedure of merging DYD and CDYD. This will be used only for couples of intersecting ones, but can be defined regardless of whether they intersect. The operation is illustrated in Figure 4.
For any two DYD, and , the minimal DYD containing is well defined. We denote it by and call it their span. The operation is associative666Associativity was referred to as commutativity by previous authors . and commutative.
Let be defined by (see (8)) and similarly for . Let be the vector such that for . Let be the set of such that for all with we have . We denote by the DYD defined by and claim that for any DYD we have , which is enough to conclude that is well defined. Let be defined by .
Note that for each (and in fact in ) there is a sequence of points in or converging to , so that (by extraction of a subsequence) there exists with . Similarly, there is a sequence of points in or converging to the boundary of , so that and similarly for . Thus, we do have .
Finally, the commutativity is obvious and the associativity follows from the characterisation of as the minimal DYD containing both and . ∎
We analogously define the span