One-dimensional long-range percolation: a numerical study

TL;DR

This study investigates bond percolation on a one-dimensional chain with power-law bond probabilities, introducing an efficient Monte Carlo algorithm to determine critical points and exponents, and compares results with theoretical predictions.

Contribution

The paper presents a new N-order Monte Carlo algorithm for long-range percolation and provides numerical estimates of critical values and exponents across different sigma values.

Findings

Critical percolation threshold C_c as a function of sigma.

Critical exponents match mean-field predictions within numerical precision.

No correction to the anomalous dimension eta from correlation effects.

Abstract

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability $C/ r^{1+\sigma}$, where $r$ is the distance length between distinct sites. We introduce and test an order $N$ Monte Carlo algorithm and we determine as a function of $\sigma$ the critical value $C_{c}$ at which percolation occurs. The critical exponents in the range $0<\sigma<1$ are reported and compared with mean-field and $\varepsilon$-expansion results. Our analysis is in agreement, up to a numerical precision $\approx 10^{-3}$, with the mean field result for the anomalous dimension $\eta=2-\sigma$, showing that there is no correction to $\eta$ due to correlation effects.