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

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 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. 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 0. This constitutes an important step toward understanding the self-organized critical behavior of such processes.