Near-critical avalanches in 2D frozen percolation and forest fires

TL;DR

This paper investigates near-critical avalanches in frozen percolation and forest fire models on the triangular lattice, revealing logarithmic scaling behaviors and advancing understanding of self-organized criticality in these processes.

Contribution

It provides the first detailed analysis of avalanche sizes and counts near criticality in frozen percolation and forest fire models, extending previous work with new coupling and exploration techniques.

Findings

Number of frozen clusters around a vertex scales as (log(96/5))^{-1} log log N

Number of burnt clusters scales as (log(96/41))^{-1} log log (zeta^{-1})

Most clusters have volume approximately zeta^{-91/55}

Abstract

We study two closely related processes on the triangular lattice: frozen percolation, where connected components of occupied vertices freeze (they stop growing) as soon as they contain at least $N$ vertices, and forest fire processes, where connected components burn (they become entirely vacant) at rate $\zeta > 0$. In this paper, we prove that when the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of near-critical "avalanches". More specifically, we analyze the avalanches, all the way up to the natural characteristic scale of each model, which constitutes an important step toward understanding the self-organized critical behavior of such processes. For frozen percolation, we show in particular that the number of frozen clusters surrounding a given vertex is asymptotically equivalent to $(\log(96/5))^{-1} \log \log N$ as $N \to \infty$. A similar mechanism underlies forest fires, enabling us to obtain an analogous result for these processes, but with substantially more work: the number of burnt clusters is equivalent to $(\log(96/41))^{-1} \log \log (\zeta^{-1})$ as $\zeta \searrow 0$. Moreover, almost all of these clusters have a volume $\zeta^{- 91/55 + o(1)}$. For forest fires, the percolation process with impurities introduced in arXiv:1810.08181 plays a crucial role in our proofs, and we extend the results in that paper, up to a positive density of impurities. In addition, we develop a novel exploration procedure to couple full-plane forest fires with processes in finite but large enough (compared to the characteristic scale) domains.