About the fastest growth of Order Parameter in Models of Percolation

TL;DR

This paper proposes a simple test based on the growth exponent of the largest component to determine whether percolation transitions are continuous or discontinuous, supported by detailed finite-size scaling analysis.

Contribution

It introduces the growth exponent $oldsymbol{ extit{ extchi}}$ as a litmus test for the nature of percolation transitions and analyzes related finite-size scaling behaviors.

Findings

Growth exponent $oldsymbol{ extit{ extchi}}$ distinguishes transition types

Finite-size scaling of maximal jump distributions analyzed

Scaling laws validated for various order parameter distributions

Abstract

Can there be a `Litmus test' for determining the nature of transition in models of percolation? In this paper we argue that the answer is in the affirmative. All one needs to do is to measure the `growth exponent' $\chi$ of the largest component at the percolation threshold; $\chi < 1$ or $\chi = 1$ determines if the transition is continuous or discontinuous. We show that a related exponent $\eta = 1 - \chi$ which describes how the average maximal jump sizes in the Order Parameter decays on increasing the system size, is the single exponent that describes the finite-size scaling of a number of distributions related to the fastest growth of the Order Parameter in these problems. Excellent quality scaling analysis are presented for the two single peak distributions corresponding to the Order Parameters at the two ends of the maximal jump, the bimodal distribution constructed by interpolation of these distributions and for the distribution of the maximal jump in the Order Parameter.