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 $p\neq p_c$ to a power-law 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 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2016
- DOI:
- 10.48550/arXiv.1611.01147
- arXiv:
- arXiv:1611.01147
- Bibcode:
- 2016arXiv161101147G
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics;
- 60K35;
- 82B20;
- 82B27;
- 82C20
- E-Print:
- 39 pages, 8 figures