Finding critical points and correlation length exponents using finite size scaling of Gini index

TL;DR

This paper introduces a finite size scaling method using the Gini index to accurately determine critical points and correlation length exponents in phase transitions, validated through simulations of various models.

Contribution

It proposes a novel scaling approach based on the Gini index to find critical points and exponents, applicable across different systems.

Findings

Gini index variability is system size independent near criticality.

Scaling form for Gini index is validated in multiple models.

Method accurately estimates critical points and exponents.

Abstract

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through Gini index ($g$), therefore, leads to a scaling form $g=G\left[|F-F_c|N^{1/d\nu}\right]$, where $F$ denotes the driving parameter for the transition (e.g., temperature $T$ for ferromagnetic to paramagnetic transition transition, or lattice occupation probability $p$), $N$ is the system size, $d$ is the spatial dimension and $\nu$ is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice and the fiber bundle model of fracture.