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

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.

💡 Research Summary

The paper introduces an extensive family of previously unstudied geometric critical exponents for the random‑cluster model (RCM) and establishes rigorous scaling relations that connect these new exponents to the well‑known k‑arm exponents. The authors begin by recalling that the RCM unifies bond percolation (q = 1) and the q‑state Potts model (q = 2) within a single probabilistic framework, with the cluster weight q related to the Coulomb‑gas coupling g by q = 2 + 2 cos(gπ/2) for 2 ≤ g ≤ 4.

They then define a set of geometric observables: the average shortest‑path length between two points in a cluster, the fractal dimension of the cluster hull, and various ratios of cluster volume to surface area. Each observable is indexed by an integer n, yielding new exponents α_n, β_n, γ_n, etc. By invoking standard scaling hypotheses, they derive linear relations between these new exponents and the traditional k‑arm exponents π_k. For example, α_n =