Convergence of threshold estimates for two-dimensional percolation

TL;DR

This paper investigates how different estimates of the percolation threshold for a 2D square lattice converge as system size increases, using a new simulation algorithm, and measures the correction-to-scaling exponent for the average-probability estimate.

Contribution

It introduces a new simulation algorithm for microcanonical samples and analyzes the convergence behavior of various percolation threshold estimates, including measuring the correction-to-scaling exponent.

Findings

Average-probability estimate convergence follows a non-trivial correction-to-scaling exponent of 0.90(2).

Median and cell-to-cell estimates converge faster with a trivial analytic exponent.

The new algorithm effectively simulates percolation in fixed-occupancy samples.

Abstract

Using a recently introduced algorithm for simulating percolation in microcanonical (fixed-occupancy) samples, we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary, specifically for site percolation on a square lattice. We show that the convergence of the so-called "average-probability" estimate is described by a non-trivial correction-to-scaling exponent as predicted previously, and measure the value of this exponent to be 0.90(2). For the "median" and "cell-to-cell" estimates of the percolation threshold we verify that convergence does not depend on this exponent, having instead a slightly faster convergence with a trivial analytic leading exponent.