# Bootstrap Percolation and Cellular Automata

We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics. We characterize the monotone automata that almost surely fill the space starting from any nontrivial Bernoulli measure. In contrast, we show the problem is undecidable if the monotonicity condition is dropped. We also construct examples where the space-filling property depends on the initial Bernoulli measure in a non-monotone way.

## Authors

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06/15/2021

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## 1 Introduction

Bootstrap percolation is a class of deterministic growth models in random environments. The basic premise is that we have a discrete universe of sites, typically arranged on a regular lattice such as , a random subset of which are initially infected

. A deterministic rule, typically uniform in space and time, allows the infection to spread into healthy sites that have enough infected neighbors. The main quantities of interest are then the probability of every site being eventually infected (called

percolation), and the distribution of the time of infection, as a function of the initial distribution of infected sites. Bootstrap percolation was introduced by Chalupa, Leath and Reich in [4] as a model of impurities in magnetic materials. See [5] for an overview of subsequent literature.

Bootstrap percolation processes can be formalized as cellular automata (CA for short) on the binary state set that are monotone ( implies ) and freezing ( always holds) with respect to the cellwise partial order. The automaton is initialized on a random configuration , and the freezing property guarantees that the iterates converge to a limit configuration. Percolation corresponds to this limit being the all-1 configuration. We say that trivializes the initial probability measure, if percolation happens almost surely.

Dropping the monotonicity requirement results in a richer set of possible asymptotic behaviors. Such automata may still be understood as models of physical or sociological phenomena. For example, if the cells of a graph represent agents with political leanings, then non-monotone rules can model individuals becoming suspicious of a sudden influx of opposing views among their peers. Examples of freezing non-monotone CA have been considered in the literature, like the “rule one” of S. Ulam [16] as an attempt to study models of crystal growth, or “life without death” [9] which is a freezing version of Conway’s Life. The dynamics of freezing cellular automata have been studied explicitly in e.g. [7, 6, 2]. We note that in the literature it is common to require freezing CA to be decreasing rather than increasing, but here we choose to follow the opposite convention of percolation theory.

In this article we study the variety of asymptotic behaviors exhibited by monotone and non-monotone freezing CA when initialized on Bernoulli random configurations. In the monotone case, we provide a characterization of those automata under which almost all initial configurations percolate with respect to at least one nontrivial Bernoulli measure. We state the characterization in terms of two criticality classifications of bootstrap percolation models. In [8] Gravner and Griffeath study threshold growth dynamics, which are a class of binary freezing monotone CA rules defined by a fixed neighborhood and a threshold , where the local rule turns a into a precisely when the number of -states in the neighborhood is at least . They call such a CA subcritical if it has a fixed point with a nonzero but finite number of -states, supercritical if there is a configuration with a finite number of -states and contains infinitely many -states, and critical if neither condition holds.

Bollobás, Smith and Uzzell provide in [3] an a priori different classification for the dynamics of arbitrary binary freezing monotone -CA in terms of stable directions. Their definition of supercriticality agrees with that of Gravner and Griffeath, but their version of subcriticality is strictly weaker. By [3, 1], a binary freezing monotone -CA trivializes every nontrivial Bernoulli measure if and only if it is critical or supercritical and the sense of [3], and the property is decidable. Our results in Section 4 concern the dual problem: given a binary freezing monotone CA, does there exist a nontrivial Bernoulli measure it trivializes? We show that this is equivalent to the stronger version of subcriticality defined by Gravner and Griffeath. In particular, the property is also decidable. As part of our proof, we give a characterization of subcriticality using stable directions, which was stated in [3] without proof.

In Section 5 we study the larger class of binary freezing CA that may not be monotone. Our results in this context have a different flavor, as they highlight the increase in complexity of the asymptotic dynamics that results from discarding the monotonicity constraint. First, we show that while the property of not trivializing any nontrivial Bernoulli measure is still equivalent to subcriticality, it is no longer decidable. Second, we show that the measure trivialization property may be non-monotone, in the sense that there exists a freezing CA that trivializes the Bernoulli measure of weight but not the one of weight , for some

. This can be interpreted as the system having at least two phase transitions.

Several open problems arise naturally from our investigation. First, in the context of cellular automata it is natural to ask whether the results extend to arbitrary finite state sets. We prove some of our auxiliary results in this context, but our main results concern the binary case. Do monotone freezing CA with three or more states have significantly more complex dynamics than binary CA? In particular, are the analogous trivialization properties decidable? Second, our example of a freezing CA with two phase transitions can likely be generalized to realize a wide range of exotic trivialization phenomena.

## 2 Definitions

For a finite alphabet and (we will mostly be dealing with the case ), the -dimensional full shift if the set equipped with the prodiscrete topology. Elements of are called configurations. For , the -uniform configuration is defined by for all . We have an action of the additive group by homeomorphisms, called the shift action, given by . If is a poset (partially ordered set), then we see as a poset with the cellwise order: means for all .

A -dimensional pattern is a function with finite. The topology of is generated by cylinder sets of the form for a pattern . If is omitted, it is assumed to be . In a slight abose of notation, each symbol stands for the pattern with domain , so that .

A cellular automaton (CA for short) is a function defined by a finite neighborhood and a local rule with . If holds for all , we say is a radius for . By the Curtis-Hedlund-Lyndon theorem, CA are exactly the continuous functions from to itself that commute with the shift action.

Denote by the set of Borel probability measures on , and by the -invariant ones (which satisfy for all Borel sets ). We equip with the weak- topology, or convergence on cylinder sets. The support of is the unique smallest closed set with . We can apply a CA to a measure by .

For a probability vector

(that satisfies ), the Bernoulli or product measure is the unique Borel measure with for all patterns . If and , then is the Bernoulli measure with . For , denote by the unique measure with .

The convex hull of a set is denoted . The notation means “for all but finitely many ”, and means “exists infinitely many ”.

## 3 Freezing, monotonicity and measures

###### Definition 3.1.

Let be a finite poset. A cellular automaton on is freezing, if for all . It is monotone, if for all .

In this paper, when considering freezing cellular automata on , we always implicitly refer to the poset with elements and such that .

###### Lemma 3.2.

A cellular automaton on is freezing and monotone if and only if there exists a finite family of finite subsets of with the following property. For all , we have if and only if or there exists with .

###### Proof.

Given a freezing and monotone , choose as the family of minimal subsets of such that implies . The other direction is clear. ∎

###### Definition 3.3.

For a freezing monotone cellular automaton on , we write for the set given by Lemma 3.2. We also denote

 F(f)={N∈E(f)|→0∉CHull(N)},

and . For a finite family of incomparable subsets of , we denote by the cellular automaton defined by . If is a singleton, we may also abuse notation and write for .

###### Definition 3.4.

Let be a measure. The -limit set of a cellular automaton on is

 Ωμf=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⋃ν∈FSupp(ν),

where is the set of limit points of the sequence .

If is shift-invariant, then is the set of configurations such that no pattern occurring in satisfies . -limit sets were first defined in [13] in the shift-invariant case using this characterization. The measures that occur in our results are shift-invariant, but in some proofs we work with intermediate measures that are not.

###### Definition 3.5.

Let and let be a cellular automaton on . We say trivializes , if .

In cellular automata literature, a CA is called -nilpotent if for some unary configuration .

If is a poset with a maximal element , and is a freezing CA on that trivializes a full-support Bernoulli measure, then the limit measure must of course be concentrated on the -uniform point.

###### Lemma 3.6.

Let be a finite poset, of full support and a freezing cellular automaton on . The following conditions are equivalent:

• trivializes

• for some (unary)

• for some and -almost every , we have for all and all large enough (depending on ).

###### Proof.

Suppose that trivializes , so that for some . Since is shift-invariant, for each we have

 {σ→v(x)}=Ωσ→vμf=Ωμf={x}

and hence there exists such that is unary. Thus, for each finite pattern containing an occurrence of some we have , and for each all- pattern we have . This implies , the second item. The converse is clear, so the first two items are equivalent.

Denote by a maximal element of . Since is freezing,

 [m]→z⊆f−n([m]→z) (1)

for all and . From (1) and the full support of it follows that no other state than can be chosen as in the third item. Let

 B={x∈PZ2|∀→v∈Z2 ∀∞n∈N:fn(x)→v=m}

be the set of configurations that satisfy the condition of the third item. Consider . From (1) we have for all and , and .

Suppose that does not trivialize . Then there must be such that for all (otherwise we would have since sets are decreasing, a contradiction). By continuity of from above, we deduce , so . Therefore the third item does not hold.

Suppose then that the third item does not hold, so that . Since has positive measure, for some . For each , we then have , and in particular for some . For some this holds for infinitely many , so some limit point of satisfies . Thus does not trivialize . We have shown that the third item is equivalent to the first. ∎

We note that the first two items of Lemma 3.6 are equivalent even without the freezing hypothesis, and for the third item we only need the condition that some state is persistent, that is, implies .

###### Example 3.7.

Let , and let . Then for all if and only if there exists a path in such that , and for all . Indeed, if such a path exists, then every cell in it will always have the state , including the origin. On the other hand, if an infinite path does not exist, by Kőnig’s lemma there is a bound for the length of the paths. If the maximal length of a path starting from the origin in is , then that in is . Inductively, we see that .

Take the product measure for , and consider the probability that in a -random configuration there exists an infinite path of 0-states as above. This probability is clearly nonincreasing with respect to , and the infimum of those for which it equals 0 is , where is the critical probability of nearest-neighbor oriented site percolation on ; see [10, Section 12.8] for discussion on the analogous bond percolation model. If , then trivializes , and if , then it does not.

## 4 Freezing monotone CA

We begin our study of freezing monotone cellular automata on the binary alphabet by recalling the definitions of sub- and supercriticality from [8, 3]. Since the definitions are not equivalent, we rename them for consistency’s sake.

###### Definition 4.1.

Let be a freezing and monotone CA on . For a unit vector , write for the configuration defined by

 x→v→z={1,if →v⋅→z<0,0,otherwise.

A direction is stable for , if , and strongly stable if it is contained in an open interval of stable directions. We say is

• strongly subcritical, if every direction is stable,

• subcritical, if every open semicircle contains a strongly stable direction,

• weakly subcritical, if it subcritical but not strongly subcritical,

• supercritical, if some open semicircle contains no stable directions, and

• critical, if none of the above hold.

###### Lemma 4.2.

Let be a freezing and monotone CA on . Then the following are equivalent.

• is strongly subcritical.

• .

• There exists such that and is nonempty and finite.

The equivalence of the first two items was claimed in [3]. We include a proof for completeness, but postpone it until after a few auxiliary results. The first one allows us to separate convex compact sets with a rational line.

###### Lemma 4.3.

For any disjoint convex compact sets , there exists a closed rational half-plane such that and .

###### Proof.

By a standard hyperplane separation theorem, such as

[11, Theorem 4.4(i)], there exist and such that the two half-planes and satisfy and . We approximate by a rational vector . By continuity of the dot product, if the approximation is accurate enough, the half-plane satisfies the claim. ∎

###### Lemma 4.4.

Let be intersecting convex polygons such that for every face of , there exist strictly longer faces of parallel and antiparallel to it. Then the intersection contains a vertex of .

###### Proof.

First, cannot be contained in : to every face of with a positive x-component corresponds a face of with a larger positive x-component, and hence the sum of these x-components is larger than the width of .

If , then we are done. Otherwise some faces of and of intersect. If they are parallel, then one endpoint of lies in . Otherwise, let and be the faces of parallel and antiparallel to . Since is a face of , the intersection point of and is not in the interior of the trapezoid . Both and are strictly longer than , so one endpoint of lies in the interior of . ∎

###### Lemma 4.5.

Let be a finite set of directions, and let . Then there exists a polygon such that

• every face of is strictly longer than , and

• for each direction in , there exists faces of parallel and antiparallel to it.

###### Proof.

Let . For each , let , where , be a point of the complex unit circle such that the tangent of at , taken in the clockwise direction, is parallel to . Let

 ϵ=minv≠w∈W,b∈{0,1}|αv−αw−b|>0.

Order the increasingly as , and define and for all . Now satisfies the required conditions for large enough . ∎

###### Proof of Lemma 4.2.

Suppose first that is strongly subcritical. If there exists , then by Lemma 4.3 there is a closed rational half-plane with and . Translating so that lies on its border, we have for some . This implies , contradicting the stability of . Hence .

Suppose then that . For all , define by . For all , define . Let be the set of directions of the faces of the polygon , and . Let be given by Lemma 4.5 for the parameters and . Let . By Lemma 4.4, we have for all , and it follows that . Thus we can choose .

Suppose that the third condition holds, and let . Let be such that is minimal. For each with we then have (because and by choice of ). By monotonicity , and in particular . Since was arbitrary under condition and since is freezing, and is stable for . ∎

Next, we study infinite paths in -images of Bernoulli random configurations, where is a local transformation of .

###### Definition 4.6.

For a map , the dependence neighborhood at position is the minimal set such that is determined by , i.e. implies . We say is -dependent () if for any with it holds .

###### Definition 4.7.

For and , denote by the set of configurations such that there is an infinite path in such that , and for all . Denote its complement by . We may abbreviate and when is clear from the context.

###### Lemma 4.8.

For any , there is such that for any and for any -dependent map verifying for all it holds:

###### Proof.

Let us first consider the case where is the identity map. Let , which is the critical probability of directed percolation on the square lattice. It is a classical result from percolation theory that . By symmetry between states and , for both and hold.

The general case follows from the results of [14]. Simplifying the setting a bit to match our needs, we say a measure is -dependent if for any with

, the random variables

and are independent when is drawn from . On the other hand, we say dominates another measure if for any upper-closed measurable set (i.e. and for all implies ) we have . Then [14, Theorem 0.0] implies that for any and , if is a -dependent measure and is large enough, then dominates the product measure . Note that need not be shift-invariant.

For any -dependent map and any product measure it is the case that is -dependent. The domination result above allows to conclude since both and are upper-closed sets. ∎

We need one more geometric lemma before proving the main result of this section.

###### Lemma 4.9.

Let be a convex compact set. Then there exists such that and for any with , we have either or .

###### Proof.

Let be the rational half-plane given by Lemma 4.3 for and . By rescaling, we may assume with . Choose , so that . Then any forms a basis of with and satisfies , and we have either or . ∎

###### Theorem 4.10.

Let be a freezing monotone cellular automaton on . Then trivializes some nontrivial Bernoulli measure if and only if it is not strongly subcritical.

###### Proof.

Suppose first that is subcritical. By Lemma 4.2 and monotonicity, there exists a nonempty finite set such that implies for all and . Under any nontrivial Bernoulli measure , the event has a nonzero probability, so does not trivialize .

Suppose then that is not strongly subcritical. Let , and define . We will show that trivializes a nontrivial Bernoulli measure , and then so does . First, let be the vector given by Lemma 4.9 for the set . Since , Bezout’s identity gives another vector such that . Let , which now has determinant . By the assumption on , we have either or , and we may choose the former case by negating if necessary. We replace by the automaton , which clearly trivializes a given Bernoulli measure if and only if does.

We apply another automorphism of , a shear transformation for some large , to guarantee . Now there exist , computable from , such that if for all , then for all . Define the map by

 ϕ(x)(i,j)={1 if x(bi,aj)+→m=1 for all →m∈[0,b−1]×[0,a−1],0 otherwise.

As in Example 3.7, it follows from Kőnig’s Lemma that if then there is such that . Note that is -dependent. Note also that can be arbitrarily close to for all when is chosen large enough. We deduce from the first item of Lemma 4.8 that trivializes some Bernoulli measure with . ∎

###### Corollary 4.11.

Given a freezing and monotone cellular automaton on , it is decidable whether trivializes a nontrivial Bernoulli measure.

###### Theorem 4.12.

Let be a freezing monotone cellular automaton on . If is empty and there exist two linearly independent coordinates such that every contains either and , or and , then does not trivialize every nontrivial Bernoulli measure.

###### Proof.

The proof is the same as in the second direction of Theorem 4.10, but switching the roles of and in the reduction. ∎

## 5 Freezing non-monotone CA

The goal of this section is to study the same classical question of bootstrap percolation (equivalently -nilpotency by Lemma 3.6) but dropping the monotone assumption in the CA considered. We show that it gives rise to much more complexity through two results: an undecidability result and a construction of an example with two phase transitions.

###### Theorem 5.1.

The problem of whether a given binary freezing CA trivializes some nontrivial Bernoulli measure is undecidable.

###### Proof.

We first give a reduction from the halting problem of Turing machines to the above problem for freezing CA with arbitrary number of states. Then we show how to recode the construction with only two states.

Let be any Turing machine and let be a set of Wang tiles that simulates its valid space-time diagrams in the classical sense (see for example [15, §4]) with and representing the initial and halting states respectively. Without loss of generality concerning undecidability of the halting problem, we consider Turing machines that only use a right semi-infinite tape and that always go to the rightmost non-blank position before entering halting state. We construct a freezing CA with a single maximum state satisfying the following implications:

1. if halts starting from the empty tape, then there exists a finite “obstacle” pattern of states of which is not -uniform and that stays unchanged under the dynamics of whatever the context;

2. if does not halt, then, for any configuration such that the maximal rectangle containing the origin and without occurence of state inside is finite, for some finite .

In the first case, does not trivialize any Bernoulli measure, because a cell has a positive probability to be in state in the initial configuration (and then will never change) and also has a positive probability to be in a state different from inside the finite “obstacle” pattern (and will never change either). In the second case, -almost all configurations have the described property for any nontrivial Bernoulli measure , because there are almost surely occurrences of on each axis, both for positive and negative coordinates. Hence trivializes all nontrivial Bernoulli measures in this case.

We now describe the CA . Its dynamics is basically to check that the configuration belongs to some SFT of “valid” configurations, to turn any cell into when a local error is detected and to propagate the state to neighboring cells according to certain conditions.

Denote . We interpret is as a set of cardinal and diagonal directions (north, north-east, east, etc). The state set of is . We call a frame any rectangular pattern whose perimeter is made with states from in the following way: the north, east, south, west sides are respectively in state , , , , and the north-east, south-east, south-west, north-west corners are respectively in states , , , . We say that a frame contains a valid halting computation if (see Figure 1):

• the bottom-left cell of the interior is and the bottom line of the interior represents an empty tape;

• the upper-right position of the interior is ;

• any other position of the interior is in and they respect the tiling conditions of , and the constraint that the head must not escape the frame.

Note in particular that our assumption on Turing machines ensures that a halting computation always fits inside a rectangular space-time diagram with initial state at the lower-left corner and halting state at the upper-right one.

Then, valid configurations are those made of frames containing a valid halting computation, and with only state outside frames. All these conditions can be defined by a set a valid patterns (intuitively, the patterns appearing in Figure 1).

Depending of its state, we define the active neighbors of a cell:

• a cell in a state from has four active neighbors, one in each cardinal direction;

• a cell in a state has as active neighbors in the cardinal directions which do not appear in : e.g. a cell in state has active neighbours to the south and west.

The dynamics of is precisely the following:

1. if a cell belongs to some invalid pattern, then it becomes ;

2. if a cell has an occurrence of among its active neighbors, then it becomes ;

3. otherwise the state doesn’t change.

The CA has neighborhood and its local rule can be algorithmically determined from .

From this definition, it is clear that when halts starting from the empty tape, then admits a finite obstacle pattern: a frame encoding a valid computation and surrounded by state like in Figure 1. Now suppose that does not halt and consider any configuration with an occurrence of on each semi-axis, both for positive and negative coordinates.

We want to show that for some . Suppose that and consider the (finite) maximal rectangle whose interior contains the origin but no occurrence of state in . We proceed by induction on the size of . If is , then the origin is surrounded by -cells and becomes in one step.

Suppose then that the claim holds for all strictly smaller rectangles. Each side of has a neighbor outside of in state (if not would not be maximal). Hence, if is not a valid finite frame encoding a halting configuration then in one step at least one cell in the interior of becomes (either because the SFT condition is locally violated somewhere or because some cell has an active neighbor in state ). Since does not halt, this means that some appears in the interior , and in we either have the cell at the origin in state , or a smaller maximal rectangle of non- states so we can conclude by induction.

So far we have proved the undecidability result for freezing CA with an arbitrary number of states. We now adapt the construction to binary freezing CA. Precisely, for any we recursively construct a freezing CA with only two states satisfying the following:

1. if halts starting from the empty tape, then there exists a finite “obstacle” pattern of states of which is not -uniform and that stays unchanged under the dynamics of whatever the context;

2. if does not halt, then there exist a constant such that for any configuration such that the maximal rectangle containing the origin and without occurence of an block of s inside is bounded, we have for some .

For the same reasons as before, a construction with the above properties proves the undecidability result claimed in the statement of the theorem.

The CA is constructed from by a standard block encoding. Let be large enough to recode any state of as a block of s and s in the following way:

• is coded by a block of s;

• any state in is coded by an block made of an outer annulus of s, an inner annulus of s, and inside them a uniquely defined pattern of s and s.

A given block over alphabet is called valid if it is one of the coding blocks above, and invalid otherwise. The dynamics of is the following:

• if a cell is not inside the central block of some pattern of nine valid blocks, then it turns into ;

• otherwise, if the local rule of applied to the pattern encoded by the blocks yields , then it turns into ;

• otherwise the cell retains its state.

Note that by choice of the coding (frame of s inside a frame of s) there is always at most one way to find a valid block around a cell which is in state , hence the second case of the dynamics above is well-defined. Moreover, if a cell in state inside a valid block turns into , then the entire block turns into . Finally, by construction, on properly encoded configurations, exactly simulates .

Therefore, if halts, then the block encoding of a valid frame of (containing a halting computation) surrounded by blocks of s clearly forms a finite obstacle pattern under the dynamics of . Now suppose that does not halt and consider a configuration with a finite maximal rectangle around the origin not containing any block of s. Like for we proceed by induction on the size of . If doesn’t contain any valid block then every cell it contains turns into in one step and we are done. If contains a valid block with an invalid neighborhood, it turns into a