Information Percolation and Cutoff for the Random-Cluster Model

TL;DR

This paper proves that the FK-dynamics for the Random-Cluster model on a torus exhibits cutoff at a specific mixing time for small enough parameters, extending the information percolation method to this model.

Contribution

It extends the information percolation framework to analyze cutoff phenomena in the Random-Cluster model's FK-dynamics.

Findings

FK-dynamics exhibits cutoff at rac{1}{\u03bb_ty}\u221a{ ext{log} n}

Mixing time window is O(rac{ ext{log} ext{log} n})

Results hold for small enough p and any q > 1

Abstract

We consider the Random-Cluster model on $(\mathbb{Z}/n\mathbb{Z})^d$ with parameters $p \in (0,1)$ and $q\ge 1$. This is a generalization of the standard bond percolation (with open probability $p$) which is biased by a factor $q$ raised to the number of connected components. We study the well known FK-dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber Heat Bath dynamics for spin systems, and prove that for all small enough $p$ (depending on the dimension) and any $q>1$, the FK-dynamics exhibits the cutoff phenomenon at $\lambda_{\infty}^{-1}\log n$ with a window size $O(\log\log n)$, where $\lambda_{\infty}$ is the large $n$ limit of the spectral gap of the process. Our proof extends the Information Percolation framework of Lubetzky and Sly [21] to the Random-Cluster model and also relies on the arguments of Blanca and Sinclair [4] who proved a sharp $O(\log n)$ mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.