Quasi-polynomial mixing of critical 2D random cluster models

Abstract

We study the Glauber dynamics for the random cluster (FK) model on the torus $(\mathbb{Z}/n\mathbb{Z}{)}^2$ with parameters $(p,q)$, for $q\in (1,4]$ and $p$ the critical point $p_c$. The dynamics is believed to undergo a critical slowdown, with its continuous-time mixing time transitioning from $O(\log n)$ for $p\ne p_c$ to a power-law in $n$ at $p=p_c$. This was verified at $p\ne p_c$ by Blanca and Sinclair, whereas at the critical $p=p_c$, with the exception of the special integer points $q=2,3,4$ (where the model corresponds to the Ising/Potts models) the best-known upper bound on mixing was exponential in $n$. Here we prove an upper bound of $n^{O(\log n)}$ at $p=p_c$ for all $q\in (1,4]$, where a key ingredient is bounding the number of nested long-range crossings at criticality.

39 pages, 8 figures

References in corpus (1)

• Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

Cited by in corpus (3)

• The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions
• Information Percolation and Cutoff for the Random-Cluster Model
• Tunneling behavior of Ising and Potts models in the low-temperature regime