A 2D forest ﬁre process beyond the critical time Jacob van den Berg Pierre Nolin Abstract

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A 2D forest ﬁre process beyond the critical time

Jacob van den Berg∗

, Pierre Nolin†

Abstract

We study forest ﬁre processes in two dimensions. On a given planar lattice, vertices in-

dependently switch from vacant to occupied at rate 1(initially they are all vacant), and any

connected component “is burnt” (its vertices become instantaneously vacant) as soon as its car-

dinality crosses a (typically large) threshold N, the parameter of the model. This process was

considered by Brouwer and the ﬁrst author in [6].

Our analysis provides a detailed description, as N→ ∞, of the process near and beyond the

critical time tc(at which an inﬁnite cluster would arise in the absence of ﬁres). In particular we

prove a somewhat counterintuitive result: there exists d>0such that with high probability, the

origin does not burn before time tc+d. This provides a negative answer to Open Problem 4.1

of [6]. Informally speaking, the result can be explained in terms of the emergence of ﬁre lanes,

whose total density is negligible (as N→ ∞), but which nevertheless are suﬃciently robust with

respect to recoveries. We expect that such a behavior also holds for the classical Drossel-Schwabl

model.

A large part of this paper is devoted to understanding the role played by recoveries during

the time interval [tc, tc+d]. These recoveries do have a “microscopic” eﬀect everywhere on the

lattice, but it turns out that their combined inﬂuence on macroscopic scales (and in fact on

relevant “mesoscopic” scales) vanishes as N→ ∞.

In order to prove this, we use key ideas from a paper by Kiss, Manolescu and Sidoravicius

[23], introducing a suitable induction argument to extend and strengthen their results. We

then use it to prove that a deconcentration result in our earlier joint work with Kiss [8] on

volume-frozen percolation (a model without recoveries) also holds for the forest ﬁre process. As

we explain, signiﬁcant additional diﬃculties arise here, since recoveries destroy the nice spatial

Markov property of frozen percolation.

Key words and phrases: near-critical percolation, forest ﬁres, self-organized criticality.

Contents

1 Introduction 2

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Mainresults......................................... 5

1.3 Additional remarks on the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Eﬀect of recoveries: self-destructive percolation . . . . . . . . . . . . . . . . . 6

1.3.2 Eﬀect of the boundary rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

∗CWI and VU University Amsterdam; E-mail: J.van.den.Berg@cwi.nl.

†City University of Hong Kong; E-mail: bpmnolin@cityu.edu.hk. Partially supported by a GRF grant from the

Research Grants Council of the Hong Kong SAR (project CityU11307320).

arXiv:2210.05642v1 [math.PR] 11 Oct 2022

1.4 Relatedworks........................................ 8

1.4.1 Drossel-Schwabl process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Other literature on these and related subjects . . . . . . . . . . . . . . . . . . 9

1.5 Organizationofthepaper ................................. 9

2 Preliminaries: 2D percolation near criticality 10

2.1 Bernoulli site percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Behavior at and near criticality: main results . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Further results: volumes of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Geometric considerations on percolation holes 14

3.1 Deﬁnitions: weak and strong holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Geometry of holes: bottlenecks and approximability . . . . . . . . . . . . . . . . . . 16

3.3 Comments on some geometric issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Frozen percolation and forest ﬁres 18

4.1 Deﬁnitionoftheprocesses................................. 18

4.2 Successiveburnings..................................... 20

5 Positive recovery time for forest ﬁres 21

5.1 Stability for six-arm events with passage sites . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Applications of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Crossingestimate.................................. 26

5.2.2 Classical consequences of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Generalizations ....................................... 29

5.3.1 Burning the cluster of an arbitrary circuit . . . . . . . . . . . . . . . . . . . . 29

5.3.2 Near-critical extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3.3 Geodesics ...................................... 31

6 Evolution of the percolation holes 33

6.1 Robustness through recoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 Persistent barriers in forest ﬁres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7 Deconcentration for forest ﬁres and proof of main results 37

7.1 Iteration procedure and deconcentration in a ﬁnite domain . . . . . . . . . . . . . . . 37

7.2 Comparison to the full lattice and completion of the proof . . . . . . . . . . . . . . . 41

7.3 Brief discussion on frozen percolation with modiﬁed boundary rules . . . . . . . . . . 48

A Appendix: uniform approximability of percolation holes 48

1 Introduction

1.1 Background and motivation

Consider the following process on a two-dimensional lattice G= (V, E), such as the square lattice

Z2or the triangular lattice T. Initially, at time t= 0, all vertices are vacant, and they become

occupied (by a “tree”) at rate 1(the birth rate), forming occupied connected components (clusters)

on G. Additionally, each occupied vertex is “hit by lightning” at rate ζ, the parameter of the model:

when this occurs, the entire occupied cluster of the vertex “burns instantaneously” i.e. all vertices in

that cluster become vacant at once. All these burnt vertices then become occupied again (“recover”)

at rate 1, and so on.

This model is a (continuous-time) version of the Drossel-Schwabl forest ﬁre model [15], which has

received much attention in the physics literature, as well as in papers on ecology and related ﬁelds

(this classical process is often viewed as a paradigmatic situation where self-organized criticality can

be observed). In this paper we simply call it the Drossel-Schwabl model. The ﬁrst mathematically

rigorous paper where this process was studied in dimension >1is, as far as we know, the paper [6]

by one of us and Brouwer.

Note that if all ignitions are ignored, i.e. in the corresponding “birth process”, then at each time

twe have a conﬁguration where the sites of the lattice are, independently of each other, occupied

with probability p=p(t) := 1 −e−t, and vacant with probability 1−p. The study of the size of

occupied connected components (and other connectivity properties) in such random conﬁgurations

is the subject of (Bernoulli) percolation theory, a ﬁeld which has witnessed impressive developments

in the last decades (see Section 2 for a brief introduction to percolation theory, and a list of results

that will be used in this paper).

One of the ﬁrst results in percolation theory was the existence of a critical value pc, strictly

between 0and 1, such that for p>pcthere is (almost surely) an inﬁnite occupied cluster, while for

p<pcthere are only ﬁnite occupied clusters. Related to this early percolation result, the above

mentioned paper [6] raised a fundamental open problem on forest ﬁres, where tcis the time at which

an inﬁnite cluster “starts to form” in the birth process, i.e. deﬁned by the relation 1−e−tc=pc.

This open problem is essentially the following: is it true that, for all t>tc, the probability that a

given vertex, say 0, burns before time tdoes not tend to 0as ζ&0?

The following intuitive argument “by contradiction” (taken roughly from [6]) suggests an aﬃr-

mative answer to the above question:

“Suppose there is a t>tcfor which the above mentioned probability does go to 0. Take

at0∈(tc, t). Trivially the mentioned probability also tends to 0for t0. So, roughly

speaking, as ζ&0, the system at time t0looks like ordinary Bernoulli percolation with

parameter 1−e−t0. Since this is larger than pc, there is a positive probability that 0

belongs to an inﬁnite occupied cluster at time t0. However, such a cluster would burn

immediately, hence before time t: contradiction.”

As said in [6], this type of reasoning is very shaky, but its conclusion turns out to be correct for

the directed binary tree (see Lemma 4.5 in that paper). The same paper also stated a percolation-

like conjecture (Conjecture 2.1 there). This conjecture says, roughly speaking, that, for Bernoulli

percolation at p=pc, if certain macroscopic occupied clusters in an nby nbox are destroyed (made

vacant), after which every vacant vertex in the box gets, independently, an extra chance δto become

occupied, then this δneeds to be bounded away from 0as n→ ∞ to assure large-scale connectivity

in the resulting conﬁguration.

The paper also introduced a variant of the Drossel-Schwabl model where the ignition mechanism

is not Poissonian, but where instead an occupied cluster is burnt immediately when its size (volume)

is at least N, which is now the parameter of the model. From now on, this process is called the

N-forest ﬁre, and we denote the corresponding probability measure by PN. One may expect that for

very large N, it behaves in many respects the same as the Drossel-Schwabl model with parameter

ζ=1

N, and the paper [6] stated similar open problems and results for this process as for the

Drossel-Schwabl model. In particular, it contained this question:

Open Problem 4.1 in [6]. Is, for all t>tc,

lim sup

N→∞

PN0burns before time t>0? (1.1)

In addition, the following conditional result was proved, where we denote Bm:= [−m, m]2.

Theorem 4.2 in [6]. If the above-mentioned percolation-like Conjecture 2.1 in [6] is

true, then there exists a t>tcsuch that for all m≥1,

lim inf

N→∞ PN∃v∈Bm:vburns before time t≤1

2.(1.2)

Much of our later work in this ﬁeld was motivated by the open problem that we just stated (and

its analog for the Drossel-Schwabl model, Open Problem 1.1 in [6]). Our hope was that, by ﬁrst

studying similar problems for models with a more “sober” dynamics, in particular volume-frozen

percolation, some key features would be discovered, and that the combination of these features

would lead to a solution to Open Problem 4.1 in [6]. Volume-frozen percolation can roughly, but

not exactly (see the discussion in Section 1.3.2), be considered, interpreting freezing as burning, as

an N-forest ﬁre process without recoveries.

One of these features was a deconcentration phenomenon, proved for volume-frozen percolation

in our joint work [8] with Kiss. Let us brieﬂy and very informally describe this property. At each

time we can consider the connected component of non-frozen vertices containing 0, that we call the

“hole” or “island” of 0(note that at time 0, this is the set of all vertices). At some point it becomes

ﬁnite, but with a volume much bigger than N, and then, step by step (where each step corresponds

to a freezing event in the current island), it shrinks further. For 0itself to freeze, it “must” at some

step be in an island with a size of order, and larger than, N: after the next, and last, step, 0will

then (with high probability) either be and remain in an island of size >0but smaller than N, or

freeze itself (which typically has a very small probability if the former island has size of order N

but much bigger than N).

The deconcentration phenomenon says that the island sizes are very sensitive for disturbances

earlier in the sequence, so that the size of the one-but-last island mentioned above is highly un-

predictable: the probability that its size is larger, but not much larger, than N(and hence the

probability that 0eventually freezes), is very small (tends to 0as N→ ∞).

In the meantime, Kiss, Manolescu and Sidoravicius had obtained in [23] a clever proof of the

percolation-like conjecture of [6] (or, rather, an equally suitable version of that conjecture). Hence

the “conditional” Theorem 4.2 of [6], stated above, became an “unconditional” one, i.e. the “If”

part in that theorem could be removed. However, Open Problem 4.1 of [6] (also mentioned above)

remained open.

The percolation-like conjecture in [6], and the version proved in [23], suggest that, very roughly

speaking, certain regions that have become disconnected from each other by “barriers” of burnt

vertices, caused by a ﬁre near (or beyond) time tc, remain with high probability disconnected

during a time d, for some d>0which does not depend on N: the recovery of barriers is slow. This

gives hope that the islands of 0, that we alluded to earlier, remain suﬃciently intact if recoveries

are allowed, so that the deconcentration result for N-volume frozen percolation should also hold for

the N-forest ﬁre model.

In the present paper we show that this hope is indeed justiﬁed: the sequence of islands for

the N-forest ﬁre, and that for volume-frozen percolation (or, rather, the last “many” steps of these

two sequences), can be compared suﬃciently well, even though the spatial Markov property which

holds for the sequence of islands in the frozen percolation model is no longer valid in the N-forest

ﬁre. This is achieved by adapting the reasonings in Sections 6 and 7 of [8], and by developing

and applying suitable versions of the results in [23]. We thus prove that the N-forest ﬁre process

satisﬁes a similar deconcentration phenomenon as N-frozen percolation, which leads to our main

result, Theorem 1.1 below, giving a negative answer to Open Problem 4.1 of [6].

1.2 Main results

Our analysis provides a detailed understanding of the asymptotic behavior of the N-forest ﬁre

process on a time interval [0, tc+d], for some positive number dwhich does not depend on N.

It can be summarized in particular by the following result. For technical reasons explained in

Remark 2.1 below, we need to focus on the case where Gis the triangular lattice (even though we

expect the same behavior to occur on other natural two-dimensional lattices such as Z2).

Theorem 1.1. Consider the N-forest ﬁre process on the triangular lattice T, and the associated

critical time tc=tc(T). There exists d>0(universal) such that

PN0burns before time tc+d−→

N→∞ 0.(1.3)

Hence,

PN0is occupied at time t−→

N→∞ p(t)=1−e−t(1.4)

uniformly over t∈[0, tc+d]. More generally, it implies immediately that for every K≥0,

PN∃v∈BK:vburns before time tc+d−→

N→∞ 0.(1.5)

The local limit of the N-forest ﬁre process is thus simply Bernoulli site percolation with parameter

p(t).

Theorem 1.1 provides a negative answer to Open Problem 4.1 of [6] and, as pointed out in

Section 1.1, may look counterintuitive since p(t)> pcfor t∈(tc, tc+d](in other words, the density

of occupied sites is “supercritical”). The result (1.5) substantially improves Theorem 7 of [23], which

states that the liminf of the l.h.s. is ≤1

Remark 1.2. We believe that our proofs could provide an explicit upper bound in (1.3). Such a

quantitative statement would allow us to reach the same conclusion as in (1.5), while letting K→ ∞

as a (very slow) function of N.

We can also consider the N-forest ﬁre process in a ﬁnite subgraph Gof T, for which we use the

notation P(G)

N. A result similar to Theorem 1.1 holds for sequences of such domains, provided they

grow fast enough in N.

Theorem 1.3. For some d>0, the following holds. For all ε > 0, there exists η=η(ε)>0such

that: if m(N)≥N48

91 −ηfor all suﬃciently large N, then

lim sup

N→∞

P(Bm(N))

N0burns before time tc+d≤ε.

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A2DforestreprocessbeyondthecriticaltimeJacobvandenBerg*,PierreNolinAbstractWestudyforestreprocessesintwodimensions.Onagivenplanarlattice,verticesin-dependentlyswitchfromvacanttooccupiedatrate1(initiallytheyareallvacant),andanyconnectedcomponentisburnt(itsverticesbecomeinstantaneouslyvacant)asso...

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A 2D forest ﬁre process beyond the critical time Jacob van den Berg Pierre Nolin Abstract

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