Renormalization of crossing probabilities in the 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.

Figures (12)

• Figure 1: A depiction of percolation over faces of the hexagonal lattice. With probability $p$, a face of the hexagonal lattice is colored black and otherwise white with probability $1-p$. To quantify connectivity properties of the underlying lattice, one is interested in constructing paths of neighboring faces which are all colored black.
• Figure 2: A depiction of six percolation configurations corresponding to percolation over faces of the triangular lattice. Macroscopic crossings are obtained either from one side of the domain to another, or between several other possible collections of faces. Faces across which percolation occurs are colored in blue.
• Figure 3: Crossing probabilities across the triangular lattice with blue connected components can be constructed in several possible configuations within hexagons, as shown above.
• Figure 4: Disjoint connected components of the high-temperature expansion of the Loop $O(n)$ model, as depicted above, can be used to construct symmetric domains.
• Figure 5: Besides connected components displayed in Figure 4, arguments for demonstrating that crossings across symmetric domains occur with high probability are first shown to hold within the interior of a box over $\textbf{H}$.