Method for estimating critical exponents in percolation processes with low sampling

TL;DR

This paper presents a new method for accurately estimating critical points and exponents in percolation processes using low sampling, demonstrated on 2D lattices and ER networks.

Contribution

A novel approach enabling simultaneous estimation of critical parameters with limited realizations in percolation models.

Findings

Accurate critical point and exponent estimates with only a few hundred realizations.

Method applicable to explosive bond percolation on 2D lattices and ER networks.

Guidelines provided for extending the method to other systems.

Abstract

In phase transition phenomena, the estimation of the critical point is crucial for the calculation of the various critical exponents and the determination of the universality class they belong to. However, this is not an easy task, since a huge amount of realizations is needed to eliminate the noise in the data. In this paper, we introduce a novel method for the simultaneous estimation of the critical point $p_c$ and the critical exponent $\beta/\nu$, applied for the case of "explosive" bond percolation on $2D$ square lattices and ER networks. The results show that with only a few hundred of realizations, it is possible to acquire accurate values for these quantities. Guidelines are given at the end for the applicability of the method to other cases as well.