Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process

TL;DR

This paper investigates the behavior of crossing probabilities and phase transitions in Mandelbrot's fractal percolation process across different dimensions, revealing discontinuities and convergence properties at critical thresholds as the subdivision parameter grows large.

Contribution

It proves that crossing probabilities tend to one at critical values for large N, establishes discontinuity of these probabilities at critical points, and analyzes the convergence rate of critical thresholds in fractal percolation.

Findings

Crossing probability tends to one at critical threshold as N increases.

Discontinuity in crossing probability at critical point for large N.

Convergence rate of critical threshold to standard percolation critical point in 2D.

Abstract

We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or discarding each subcube with probability $p$ or $1-p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, we consider the random fractal set produced at the path-percolation critical value $p_c(N,d)$, and show that the probability that it contains a path connecting two opposite faces of the cube $[0,1]^d$ tends to one as $N \to \infty$. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of $p$, at $p_c(N,d)$ for all $N$ sufficiently large. This had previously been proved only for $d=2$ (for any $N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N \to \infty$, to the critical density $p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $\nu$ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$, showing an interesting relation with near-critical percolation.