Some geometric critical exponents for percolation and the random-cluster model

TL;DR

This paper introduces new critical exponents for the random-cluster model, confirms their scaling relations through simulations, and proposes an exact formula for the shortest-path fractal dimension in two dimensions.

Contribution

It presents several new critical exponents for the random-cluster model, relates them to k-arm exponents, and improves Monte Carlo simulation methods.

Findings

Monte Carlo simulations confirm the predicted scaling relations.

New exponents facilitate determination of k-arm exponents.

Conjectured exact formula for the shortest-path fractal dimension d_min.

Abstract

We introduce several infinite families of new critical exponents for the random-cluster model and present scaling arguments relating them to the k-arm exponents. We then present Monte Carlo simulations confirming these predictions. These new exponents provide a convenient way to determine k-arm exponents from Monte Carlo simulations. An understanding of these exponents also leads to a radically improved implementation of the Sweeny Monte Carlo algorithm. In addition, our Monte Carlo data allow us to conjecture an exact expression for the shortest-path fractal dimension d_min in two dimensions: d_min = (g+2)(g+18)/(32g) where g is the Coulomb-gas coupling, related to the cluster fugacity q via q = 2 + 2 cos(g\pi/2) with 2 \le g \le 4.