Coexistence of spanning clusters in directed percolation

TL;DR

This paper investigates the universal probability distribution of spanning clusters in directed percolation, revealing a new critical quantity and analyzing cluster properties across dimensions.

Contribution

It introduces a universal form for the distribution of spanning clusters and identifies a new critical parameter that vanishes at the upper critical dimension.

Findings

Distribution P(n) ~ exp(-α n^2) is universal in 2D and 3D

α is a new critical quantity vanishing at the upper critical dimension

Cluster masses follow a Pearson distribution with a lower cutoff

Abstract

The probability distribution for the number of top to bottom spanning clusters in Directed percolation in two and three dimensions appears to be universal and is of the form $P(n) \sim \exp(-\alpha n^2)$. We argue that $\alpha$ is a new critical quantity vanishing at the upper critical dimension. The probability distribution of the individual masses of the spanning clusters is found to have a Pearson distribution with a lower cutoff. Various properties of the clusters are reported.