Renormalization of crossing probabilities in the dilute Potts model

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.