Universal Critical Wrapping Probabilities in the Canonical Ensemble

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.