Quasipolynomial 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 continuoustime mixing time transitioning from $O(\log n)$ for $p\neq p_c$ to a powerlaw in $n$ at $p=p_c$. This was verified at $p\neq p_c$ 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 bestknown 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 longrange crossings at criticality.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.01147
 Bibcode:
 2016arXiv161101147G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60K35;
 82B20;
 82B27;
 82C20
 EPrint:
 39 pages, 8 figures