Corner contribution to percolation cluster numbers

TL;DR

This paper investigates how corners in a boundary affect the number of clusters intersecting it in 2D critical percolation, revealing universal logarithmic corrections linked to conformal invariance and confirmed by simulations.

Contribution

It provides exact formulas for corner contributions to cluster counts in 2D critical percolation using conformal invariance and confirms them with Monte Carlo simulations.

Findings

Corner effects introduce universal logarithmic corrections.

Conformal invariance and Cardy-Peschel formula are used to derive formulas.

Results are confirmed by large-scale Monte Carlo simulations.

Abstract

We study the number of clusters in two-dimensional (2d) critical percolation, N_Gamma, which intersect a given subset of bonds, Gamma. In the simplest case, when Gamma is a simple closed curve, N_Gamma is related to the entanglement entropy of the critical diluted quantum Ising model, in which Gamma represents the boundary between the subsystem and the environment. Due to corners in Gamma there are universal logarithmic corrections to N_Gamma, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.