On the Aizenman exponent in critical percolation

TL;DR

This paper numerically investigates cluster spanning probabilities in critical percolation across dimensions two to seven, confirming Aizenman's multiplicity exponent in certain dimensions and discussing a crossover to mean-field behavior near the upper critical dimension.

Contribution

It provides numerical evidence supporting Aizenman's exponent in dimensions three to five and explores the transition to mean-field behavior at higher dimensions.

Findings

Confirmation of Aizenman's exponent in 3-5 dimensions

Identification of crossover to mean-field behavior near the upper critical dimension

Use of advanced algorithms for accurate finite-size analysis

Abstract

The probabilities of clusters spanning a hypercube of dimensions two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen--Kopelman algorithm combined with Grassberger's "go with the winner" strategy for the site percolation. We carried out a finite-size analysis of the data and found that the probabilities confirm Aizenman's proposal of the multiplicity exponent for dimensions three to five. A crossover to the mean-field behavior around the upper critical dimension is also discussed.