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

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 $π/2$.