Number of spanning clusters at the high-dimensional percolation thresholds

TL;DR

This paper develops a scaling theory for the number of spanning clusters at percolation thresholds across different dimensions, supported by numerical simulations, revealing dimension-dependent behaviors and finite-size effects.

Contribution

It introduces a new scaling framework for spanning cluster counts at criticality, including boundary condition effects and corrections, validated by extensive simulations.

Findings

Number of spanning clusters becomes independent of size for d<6

At d=6, the count varies as log L, with corrections needed

For d>6, the behavior depends on boundary conditions, ranging from L^{d-6} to L^0

Abstract

A scaling theory is used to derive the dependence of the average number <k> of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6, and vary as log L at d=6. The predictions for d>6 depend on the boundary conditions, and the results there may vary between L^{d-6} and L^0. While simulations in six dimensions are consistent with this prediction (after including corrections of order loglog L), in five dimensions the average number of spanning clusters still increases as log L even up to L = 201. However, the histogram P(k) of the spanning cluster multiplicity does scale as a function of kX(L), with X(L)=1+const/L, indicating that for sufficiently large L the average <k> will approach a finite value: a fit of the 5D multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well. Numerical simulations for d>6 and for d=4 are also presented.