The phase transitions of the random-cluster and Potts models on slabs with $q \geq 1$ are sharp

TL;DR

This paper establishes the sharpness of phase transitions in the random-cluster and Potts models on certain slab graphs, demonstrating a clear transition point with exponential decay below and infinite clusters above.

Contribution

It proves the sharpness of phase transitions for the random-cluster and Potts models on slabs with general graphs, extending previous results to broader classes of graphs and interactions.

Findings

Existence of a critical parameter p_c separating decay and percolation phases

Sharp phase transition behavior on planar lattice slabs with symmetry

Extension of results to models with long-range interactions

Abstract

We prove sharpness of the phase transition for the random-cluster model with $q \geq 1$ on graphs of the form $\mathcal{S} := \mathcal{G} \times S$, where $\mathcal{G}$ is a planar lattice with mild symmetry assumptions, and $S$ a finite graph. That is, for any such graph and any $q \geq 1$, there exists some parameter $p_c = p_c(\mathcal{S}, q)$, below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards-Sokal coupling.