Crossing bonds in the random-cluster model

TL;DR

This paper derives the scaling dimension for crossing bonds in the 2D Potts model's random-cluster representation, showing it to be irrelevant, and confirms this via numerical transfer-matrix analysis.

Contribution

It provides a Coulomb gas mapping to determine the crossing-bond scaling dimension and confirms its irrelevance through numerical finite-size-scaling analysis.

Findings

Crossing bonds are irrelevant in the 2D Potts model.

Coulomb gas mapping yields the crossing-bond scaling dimension.

Numerical analysis confirms the theoretical prediction.

Abstract

We derive the scaling dimension associated with crossing bonds in the random-cluster representation of the two-dimensional Potts model, by means of a mapping on the Coulomb gas. The scaling field associated with crossing bonds appears to be irrelevant, on the critical as well as on the tricritical branch. The latter result stands in a remarkable contrast with the existing result for the tricritical O(n) model that crossing bonds are relevant. In order to obtain independent confirmation of the Coulomb gas result for the crossing-bond exponent, we perform a finite-size-scaling analysis based on numerical transfer-matrix calculations.