Recovery from forest fires (self-destructive percolation) on $\mathbb{Z}^d$ in low dimensions

In the forest-fire model of van den Berg and Brouwer — self-destructive percolation — the infinite cluster of a supercritical Bernoulli percolation configuration on a graph is burned, after which a small density $\varepsilon$ of independently chosen new edges is added. The graph recovers from fires if for every $\varepsilon>0$ some supercritical parameter $p$ still leaves an infinite cluster after burning and $\varepsilon$-reinforcement. Ahlberg, Duminil-Copin, Kozma and Sidoravicius proved that $\mathbb{Z}^d$ recovers in sufficiently high dimensions (and on the stretched-out 7-dimensional lattice), the proof reducing to superlinear decay of the critical one-arm probability; Kiss, Manolescu and Sidoravicius proved that planar lattices do not recover, settling the original 2004 conjecture. The intervening cases are open: for the nearest-neighbour lattice it is not known whether $\mathbb{Z}^d$ recovers in any dimension $3\le d\le 10$ — in particular in $d=3,4,5,6$ — the smallest recovering dimension is unknown, and the site-percolation analogue is open even in high dimensions.

Background

Bernoulli bond percolation on the hypercubic lattice $\mathbb{Z}^d$ declares each edge open independently with probability $p$; above a critical value $p_c$ there is almost surely a unique infinite connected component of open edges (the infinite cluster $C_\infty$), while below $p_c$ all clusters are finite. A basic quantity at criticality is the one-arm probability $\mathbb{P}_{p_c}(0 \leftrightarrow \partial B(0,r))$, the probability that the origin is connected by open edges to some vertex at distance $r$; its polynomial decay rate in $r$ is the one-arm exponent.

Self-destructive percolation, introduced by Jacob van den Berg and Rachel Brouwer in 2004 as a static toy model for forest-fire processes with self-organized criticality (the phenomenon of a driven-dissipative system settling near a critical point without any tuned parameter), asks what remains of a supercritical configuration after its infinite cluster is destroyed. The construction (following the source paper) builds all configurations on one probability space: assign independent Uniform$[0,1]$ variables to the edges and let $\omega_p$ be the set of edges with value at most $p$, so that $\omega_{p_1}\subseteq\omega_{p_2}$ whenever $p_1\le p_2$. Burning turns $\omega_p$ into $\tilde\omega_p$ by closing every edge that belongs to an infinite cluster of $\omega_p$. Recovery then forms $\tilde\omega_{p,\varepsilon}(e)=\max\{\tilde\omega_p(e),\omega'_\varepsilon(e)\}$, where $\omega'_\varepsilon$ is an independent Bernoulli-$\varepsilon$ bond configuration — a small density of freshly grown connections allowed anywhere, including across the burned region. A graph $G$ recovers from fires if for every $\varepsilon>0$ there exists $p>p_c(G)$ such that $\tilde\omega_{p,\varepsilon}$ contains an infinite connected component almost surely. The question it probes: just above $p_c$ the infinite cluster occupies an arbitrarily small fraction of space, so one might expect that after burning it, an arbitrarily small reinforcement recreates an infinite cluster; whether this holds depends subtly on dimension and geometry. Replacing edges by vertices (sites declared open or closed) gives the site-percolation analogue, called site-recovery.

In the plane the expectation fails. Van den Berg and Brouwer proved that the binary tree (a non-amenable graph, meaning the boundary of a finite set of vertices is comparable in size to the set itself) site-recovers, and conjectured that $\mathbb{Z}^2$ does not site-recover. Two-dimensional partial results — box-crossing estimates (bounds on the probability that open paths cross long thin rectangles) and continuity statements, together with a linear lower bound $\delta_c(p)\ge(p-p_c)/p$ on the critical reinforcement (the minimal reinforcement density needed to recreate an infinite cluster at parameter $p$) — preceded the complete solution by Demeter Kiss, Ioan Manolescu and Vladas Sidoravicius (2015): there exists $\delta>0$ such that for every $p>p_c$ the burned-then-$\delta$-reinforced configuration has no infinite cluster, proved for site percolation on $\mathbb{Z}^2$ and stated (with the same methods) for site and bond percolation on arbitrary planar lattices with sufficient symmetry. Planarity is decisive: an infinite cluster in the plane, however thin, cuts the remaining lattice into finite pieces.

In the opposite direction, the source paper (Daniel Ahlberg, Hugo Duminil-Copin, Gady Kozma, Vladas Sidoravicius, 2013/2015) proved that $\mathbb{Z}^d$ recovers from fires for all sufficiently large $d$, and noted that the proof extends to the stretched-out seven-dimensional lattice (a spread-out lattice, where edges join vertices at $\ell_\infty$-distance in a fixed range $[1,L]$ with $L$ large). It in fact proves the stronger statement that for every $\varepsilon>0$ there is $p>p_c$ such that $\omega_{p_c+\varepsilon}\setminus C_\infty(\omega_p)$ percolates, via a two-scale renormalization: crossings of two-dimensional slabs at density $p_c+\varepsilon$ survive removal of the sparse set of vertices connected far away at criticality. The single analytic input the proof really needs is a superlinear one-arm decay, $\mathbb{P}_{p_c}(0 \leftrightarrow \partial B(0,r))\le r^{-1-c}$ for some $c>0$. This is known only where the lace expansion applies — a perturbative technique yielding mean-field behavior (critical behavior matching that of percolation on an infinite-dimensional tree, where the critical exponents take exactly known values) above the upper critical dimension $6$, the dimension above which such behavior is expected to hold — via finiteness of the triangle diagram (a certain sum over two-point connectivities) — giving the one-arm decay $r^{-2}$ in high dimensions (Gady Kozma and Asaf Nachmias), known for nearest-neighbour lattices in $d\ge 19$ at the time of the source paper and since extended to $d\ge 11$ by Robert Fitzner and Remco van der Hofstad, and for spread-out lattices in every $d>6$.

For the nearest-neighbour lattice nothing is proved in $3\le d\le 10$. In $\mathbb{Z}^3$ the one-arm probability is bounded below by $cr^{-1}$ (a classical bound recorded in the source paper), so the renormalization approach cannot work there; numerical evidence summarized by Geoffrey Grimmett, Alexander Holroyd and Gady Kozma suggests the required superlinear one-arm decay also fails in $d=4$ and first holds in $d=5$. Thus in $d=3,4$ recovery, if true, requires essentially different arguments, while in $d=5,6$ (and nearest-neighbour $d=7$–$10$) it would follow from the conjectured one-arm decay, itself a major open problem of intermediate-dimensional percolation. A revealing companion question — whether the complement of the infinite cluster contains any infinite connected component (in the ordinary graph sense, over both open and closed edges) for $p$ slightly above $p_c$ — is likewise open below the lace-expansion range: Grimmett–Holroyd–Kozma (2014) proved it for $d\ge 19$ and spread-out $d\ge 7$; Bounghun Bock, Michael Damron, C. M. Newman and Vladas Sidoravicius (2020) improved this to $d\ge 10$ by an elementary argument producing infinite paths of shielded vertices (vertices with all adjacent edges closed), and to $d\ge 7$ using simulated values of $p_c$.

Problem Statement

Let $\mathbb{Z}^d$ denote the $d$-dimensional nearest-neighbour hypercubic lattice and $p_c$ its Bernoulli bond percolation threshold. Build all percolation configurations on one probability space by assigning each edge an independent Uniform$[0,1]$ value and letting $\omega_p$ be the set of edges with value at most $p$, so that $\omega_{p_1}\subseteq\omega_{p_2}$ whenever $p_1\le p_2$. Let $\tilde\omega_p$ be obtained from $\omega_p$ by closing every edge that belongs to an infinite cluster of $\omega_p$, and let $\tilde\omega_{p,\varepsilon}(e)=\max\{\tilde\omega_p(e),\omega'_\varepsilon(e)\}$, where $\omega'_\varepsilon$ is an independent Bernoulli-$\varepsilon$ bond configuration on the same edge set. $\mathbb{Z}^d$ recovers from fires if for every $\varepsilon>0$ there exists $p>p_c$ such that $\tilde\omega_{p,\varepsilon}$ contains an infinite connected component almost surely.

Known: planar lattices ($d=2$) do not recover, in both the site and the bond version (Kiss–Manolescu–Sidoravicius); $\mathbb{Z}^d$ recovers for all sufficiently large $d$, and the stretched-out (spread-out) lattice — in which edges join vertices at $\ell_\infty$-distance in a fixed range $[1,L]$ with $L$ large — recovers in $d=7$ (Ahlberg–Duminil-Copin–Kozma–Sidoravicius).

Determine the remaining cases — the open core of the problem:

1. Does $\mathbb{Z}^d$ recover from fires for $d=5$ and $d=6$ (and for the nearest-neighbour lattice in $d=7,\dots,10$)?
2. Does $\mathbb{Z}^d$ recover from fires for $d=3$ and $d=4$, where the superlinear one-arm decay is known ($d=3$) or believed ($d=4$) to fail, so that the high-dimensional approach cannot apply and new ideas are required?
3. Determine the set $\{d\ge 1:\ \mathbb{Z}^d \text{ recovers from fires}\}$, and determine the smallest dimension in which the nearest-neighbour hypercubic lattice recovers; in particular, decide whether recovery holds in every $d\ge 3$ or fails in some intermediate dimension.
4. Does $\mathbb{Z}^d$ site-recover from fires for all sufficiently large $d$ (the same statement with site percolation in place of bond percolation)?

These components share one scientific objective: the dimension classification of recovery from fires (self-destructive percolation) on hypercubic lattices; components 1 and 2 are subsumed by component 3.

References

1. Daniel Ahlberg, Hugo Duminil-Copin, Gady Kozma, Vladas Sidoravicius, 'Seven-dimensional forest fires', Ann. Inst. Henri Poincaré Probab. Statist. 51(3) (2015) 862–866. DOI: 10.1214/13-AIHP587; arXiv:1302.6872. https://arxiv.org/abs/1302.6872
2. Demeter Kiss, Ioan Manolescu, Vladas Sidoravicius, 'Planar lattices do not recover from forest fires', Ann. Probab. 43(6) (2015) 3216–3238. DOI: 10.1214/14-AOP958; arXiv:1312.7004. https://arxiv.org/abs/1312.7004
3. Jacob van den Berg, Rachel Brouwer, 'Self-destructive percolation', Random Structures Algorithms 24(4) (2004) 480–501. DOI: 10.1002/rsa.20003. https://doi.org/10.1002/rsa.20003
4. Daniel Ahlberg, Vladas Sidoravicius, Johan Tykesson, 'Bernoulli and self-destructive percolation on non-amenable graphs', Electron. Commun. Probab. 19 (2014), paper 40, 1–6. DOI: 10.1214/ECP.V19-2611; arXiv:1302.6870. https://arxiv.org/abs/1302.6870
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9. Jacob van den Berg, Rachel Brouwer, Bálint Vágvölgyi, 'Box-crossings and continuity results for self-destructive percolation in the plane', in 'In and Out of Equilibrium 2' (Progr. Probab. 60), Birkhäuser, Basel (2008) 117–135. DOI: 10.1007/978-3-7643-8786-0_6. https://doi.org/10.1007/978-3-7643-8786-0_6
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12. Raphaël Cerf, Nicolas Forien, 'Some toy models of self-organized criticality in percolation', ALEA Lat. Am. J. Probab. Math. Stat. 19 (2022) 367–416. DOI: 10.30757/ALEA.v19-14; arXiv:1912.06639. https://arxiv.org/abs/1912.06639
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