Mapping functions and critical behavior of percolation on rectangular domains

TL;DR

This paper investigates how the probability of percolation and existence in rectangular domains depends on aspect ratio, using mapping functions to understand superscaling behavior and extract critical exponents.

Contribution

It introduces a method to analyze percolation probabilities on rectangles via mapping functions, providing a way to determine critical exponents from numerical data.

Findings

Superscaling exponents a and b are derived from mapping functions.

Mapping functions f_R and g_R approximate the superscaling behavior.

Critical exponents are obtained numerically from the mapping functions.

Abstract

The existence probability $E_p$ and the percolation probability $P$ of the bond percolation on rectangular domains with different aspect ratios $R$ are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of $E_p$ and $P$ for such systems with exponents $a$ and $b$, respectively, found by Watanabe, Yukawa, Ito, and Hu in [Phys. Rev. Lett. \textbf{93}, 190601 (2004)] can be understood from the lower order approximation of the mapping functions $f_R$ and $g_R$ for $E_p$ and $P$, respectively; the exponents $a$ and $b$ can be obtained from numerically determined mapping functions $f_R$ and $g_R$, respectively.