From discrete to continuous percolation in dimensions 3 to 7

TL;DR

This paper introduces a new method to accurately estimate percolation thresholds and critical exponents in high-dimensional aligned hypercube models by analyzing the convergence of discrete models to continuous limits.

Contribution

It establishes a universal power-law convergence with exponent 3/2 and provides improved estimates of percolation thresholds and critical exponents in dimensions 3 to 7.

Findings

Universal convergence exponent θ = 3/2 for discrete to continuous percolation

More accurate percolation thresholds in dimensions 3 to 7

Refined critical exponent ν in dimensions 4 and 5

Abstract

We propose a method of studying the continuous percolation of aligned objects as a limit of a corresponding discrete model. We show that the convergence of a discrete model to its continuous limit is controlled by a power-law dependency with a universal exponent $\theta = 3/2$. This allows us to estimate the continuous percolation thresholds in a model of aligned hypercubes in dimensions $d = 3,\ldots,7$ with accuracy far better than that attained using any other method before. We also report improved values of the correlation length critical exponent $\nu$ in dimensions $d = 4,5$ and the values of several universal wrapping probabilities for $d=4,\ldots,7$.