Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Abstract

The tricritical behavior of the two-dimensional q-state Potts model with vacancies for 0 ≤ q ≤ 4 is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The known 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 which closed graph configurations are generated by means of a Metropolis update algorithm involving single plaquettes.

Two-dimensional critical Potts and its tricritical shadow

“…This behavior is different from what we found in another numerical study of the hulls of geometrical clusters in the 2D Ising model [22]. In that Monte Carlo study, we used a plaquette update to directly simulate the hulls.…”

“…The theoretical predictions (22) can be directly verified through Monte Carlo simulations, using the Swendsen-Wang ∝ n −31/15…”

Critical loop gases and the worm algorithm

“…In this paper, which extends previous work by two of us on the subject [10,11], we describe estimators of physical observables that naturally arise in a loop gas and that allow determining the standard critical exponents. Our approach, put forward in Section 2, amalgamates concepts from percolation theory-the paradigm of a geometrical phase transition-and the theory of self-avoiding random walks.…”

Nematic-disordered phase transition in systems of long rigid rods on two-dimensional lattices

“…For instance, it is known that the exponent characterizing the decay of cluster size distribution of critical Fortuin-Kasteleyn clusters [27] in the q-state Potts model [28,29] has a non-trivial dependence on q. We denote all sites occupied by x-mers by 1 and the rest by zero.…”