Universal Critical Wrapping Probabilities in the Canonical Ensemble

TL;DR

This paper investigates how critical wrapping probabilities differ between canonical and grand-canonical ensembles in the Potts model, revealing universal but ensemble-dependent values and finite-size corrections, supported by simulations.

Contribution

It derives the universal critical wrapping probabilities in the canonical ensemble and introduces an efficient cluster algorithm for simulations.

Findings

Critical wrapping probabilities differ between ensembles for certain conditions.

Universal values are maintained in the canonical ensemble with specific modifications.

Finite-size corrections are introduced in the canonical ensemble.

Abstract

Universal dimensionless quantities, such as Binder ratios and wrapping probabilities, play an important role in the study of critical phenomena. We study the finite-size scaling behavior of the wrapping probability for the Potts model in the random-cluster representation, under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We derive that, in the limit $L \rightarrow \infty$, the critical values of the wrapping probability are different from those of the unconstrained model, i.e. the model in the grand-canonical ensemble, but still universal, for systems with $2y_t - d > 0$ where $y_t = 1/\nu$ is the thermal renormalization exponent and $d$ is the spatial dimension. Similar modifications apply to other dimensionless quantities, such as Binder ratios. For systems with $2y_t-d \le 0$, these quantities share same critical universal values in the two ensembles. It is also derived that new finite-size corrections are induced. These findings apply more generally to systems in the canonical ensemble, e.g. the dilute Potts model with a fixed total number of vacancies. Finally, we formulate an efficient cluster-type algorithm for the canonical ensemble, and confirm these predictions by extensive simulations.