Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model

TL;DR

This paper explores the relationship between geometrical and Fortuin-Kasteleyn clusters in the 2D q-state Potts model, revealing their shared universality class through a transformation that links critical and tricritical behaviors, supported by Monte Carlo simulations.

Contribution

It demonstrates a transformation connecting geometrical and FK clusters in the Potts model, preserving universality class and critical properties, supported by new Monte Carlo methods.

Findings

Geometrical and FK clusters are in the same universality class.

The transformation maps critical and tricritical regimes onto each other.

Monte Carlo simulations support the theoretical connection.

Abstract

The tricritical behavior of the two-dimensional $q$-state Potts model with vacancies for $1\leq q \leq4$ is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The close connection between the critical properties of the pure model and the tricritical properties of the diluted model is shown to be reflected in an intimate relation between Fortuin-Kasteleyn and geometrical clusters: The same transformation mapping the two critical regimes onto each other also maps the two cluster types onto each other. The map conserves the central charge, so that both cluster types are in the same universality class. The geometrical picture is supported by a Monte Carlo simulation of the high-temperature representation of the Ising model ($q=2$). In this new numerical approach, closed graph configurations are generated by means of a Metropolis update algorithm, involving single plaquettes.