Continuum percolation of overlapping discs with a distribution of radii having a power-law tail

TL;DR

This paper investigates how the distribution of disc radii with a power-law tail affects continuum percolation, revealing phase-dependent critical exponents and providing numerical and theoretical insights into the percolation threshold and correlation length.

Contribution

It introduces an analysis of continuum percolation with power-law distributed radii, identifying phase-dependent critical exponents and proposing an approximate renormalization scheme.

Findings

Power-law tail causes power-law decay of two-point function at low densities.

Critical exponents depend on the tail parameter a for certain regimes.

The percolation threshold varies with the power-law parameter a.

Abstract

We study continuum percolation problem of overlapping discs with a distribution of radii having a power-law tail; the probability that a given disc has a radius between $R$ and $R+dR$ is proportional to $R^{-(a+1)}$, where $a > 2$. We show that in the low-density non-percolating phase, the two-point function shows a power law decay with distance, even at arbitrarily low densities of the discs, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short range values for $a > 3 - \eta_{sr}$ whereas they depend on $a$ for $a < 3-\eta_{sr}$ where $\eta_{sr}$ is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, $a \leq 2$, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent $\nu$ and the percolation threshold. We carry out Monte-Carlo simulations and determine the exponent $\nu$ as a function of $a$. The determined values of $\nu$ show that it is independent of the parameter $a$ for $a>3 - \eta_{sr}$ and is equal to that for the lattice percolation problem, whereas $\nu$ varies with $a$ for $2<a<3 - \eta_{sr}$. We also determine the percolation threshold of the system as a function of the parameter $a$.