On the convergence of percolation probability functions to cumulative distribution functions on square lattices with (1,0)-neighborhood

TL;DR

This paper investigates how percolation probability functions on square lattices with beta-distributed site weights converge to the cumulative distribution functions of these weights, using Monte Carlo simulations and empirical analysis.

Contribution

It introduces empirical hypotheses linking percolation thresholds to beta-distribution quantiles and demonstrates convergence of percolation probability functions to distribution functions.

Findings

Percolation thresholds approximate specific beta-distribution quantiles.

Percolation probability functions tend to the cumulative distribution functions for supercritical probabilities.

Monte Carlo estimates effectively model percolation behavior on weighted lattices.

Abstract

We consider a percolation model on square lattices with sites weighted by beta-distributed random variables $S\sim \mathrm{Beta}(a,b)$ with a positive real parameters $a>0$ and $b>0$. Using the Monte Carlo method, we estimate the percolation probability $P_\infty$ as a relative frequency $P^*_\infty$ averaged over the target subset of sites on a square lattice. As a result of the comparative analysis, we formulate two empirical hypotheses: the first on the correspondence of percolation thresholds $p_c$ to $p_0$-quantiles (where $p_0=0.592746\ldots$) of random variables $S_i$ weighing sites of the square lattice with $(1,0)$-neighborhood, and the second on the convergence of statistical estimates of percolation probability functions $P^*_\infty(p)$ to cumulative distribution functions $F_{S_i}(p)$ of these variables $S_i$ for the supercritical values of the occupation probability $p\geq p_c$.