The Approximate Invariance of the Average Number of Connections for the Continuum Percolation of Squares at Criticality

TL;DR

This study uses Monte Carlo simulations to analyze the average number of connections at the percolation threshold for squares and sticks in 2D, revealing an approximate invariance across different orientations.

Contribution

It provides new simulation-based estimates of the average connections at criticality for squares with various orientations, highlighting the invariance of this measure.

Findings

The average number of connections varies less than 1.6% across different angular orientations.

Significant differences are found between simulations and previous analytical results for randomly oriented squares.

The results improve understanding of percolation thresholds in 2D systems of anisotropic objects.

Abstract

We perform Monte Carlo simulations to determine the average excluded area $<A_{ex}>$ of randomly oriented squares, randomly oriented widthless sticks and aligned squares in two dimensions. We find significant differences between our results for randomly oriented squares and previous analytical results for the same. The sources of these differences are explained. Using our results for $<A_{ex}>$ and Monte Carlo simulation results for the percolation threshold, we estimate the mean number of connections per object $B_c$ at the percolation threshold for squares in 2-D. We study systems of squares that are allowed random orientations within a specified angular interval. Our simulations show that the variation in $B_c$ is within 1.6% when the angular interval is varied from 0 to $\pi/2$.