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

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 $P_{\infty }^{\ast }$ 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 $p_0=0.592746\dots$) 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 $P_{\infty }^{\ast }(p)$ to cumulative distribution functions $F_{S_i}(p)$ of these variables $S_i$ for the supercritical values of the occupation probability $p\ge p_c$.

12 pages, 4 figures