Renormalization of crossing probabilities in the dilute Potts model

TL;DR

This paper extends novel renormalization techniques for crossing probabilities from the Random-Cluster model to the dilute Potts model, analyzing its behavior through a quadrichotomy at different phases using a spin representation derived from the loop $O(n)$ model.

Contribution

It introduces and studies new renormalization arguments for crossing probabilities in the dilute Potts model, expanding their applicability beyond self-dual models.

Findings

Characterizes four possible behaviors of the dilute Potts model at different phases.

Adapts renormalization methods to non-self-dual models.

Connects high-temperature expansion of loop $O(n)$ measure to dilute Potts model.

Abstract

A recent paper due to Duminil-Copin and Tassion from $2019$ introduces a novel argument for obtaining estimates on horizontal crossing probabilities of the Random-Cluster model, in which a range of four possible behaviors, through a quadrichotomy, is established. Novel renormalization arguments for crossing probabilities that the authors propose are studied in other models of interest that are not self-dual, specifically for the dilute Potts model. The probability measure of this model, through a suitably defined spin representation, is obtained from the high-temperature expansion of the loop $O(n)$ measure. The dilute Potts model was originally introduced in $1991$ by Nienhuis and is another model whose possible range of behaviors can be analyzed through a quadrichotomy; the range of four possible behaviors of the model can be respectively characterized at subcriticality, or supercriticality, in addition to either being critical discontinuous, or critical continuous. The exponential factor that is inserted into the Loop $O(n)$ model to quantify properties of the high-temperature phase is proportional to the summation over all spins, and the number of monochromatically colored triangles over a finite volume, which is in exact correspondence with the parameter of a Boltzmann weight introduced in Nienhuis' paper detailing extensions of the $q$-state Potts model.