2021/06/18 by Wai‐Kit Lam, Lam, Wai-Kit, Pierre Nolin +1
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2106.10183
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 ζ> 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 → ∞. 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 (ζ-1) as ζ\searrow 0. Moreover, almost all of these clusters have a volume ζ- 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.