A coarse graining for the Fortuin-Kasteleyn measure in random media

TL;DR

This paper uses multi-scale analysis to describe the typical geometric structure of clusters in the Fortuin-Kasteleyn measure within random media, applicable in dimensions two and higher under certain percolation conditions.

Contribution

It extends previous work by Pisztora, providing a new coarse-graining method for FK measures in disordered media, aiding the analysis of supercritical regimes.

Findings

Cluster structures are characterized in high dimensions under slab percolation.

The results apply to disordered FK, Ising, and Potts models in the supercritical phase.

Provides a new analytical tool for disordered statistical mechanics models.

Abstract

By means of a multi-scale analysis we describe the typical geometrical structure of the clusters under the FK measure in random media. Our result holds in any dimension greater or equal to 2 provided that slab percolation occurs under the averaged measure, which should be the case in the whole supercritical phase. This work extends the one of Pisztora and provides an essential tool for the analysis of the supercritical regime in disordered FK models and in the corresponding disordered Ising and Potts models.