UC Berkeley

The random-cluster model has been widely studied as a unifying framework for spin systems, random graphs and electrical networks. The random-cluster model on a graph~$G=(V,E)$ with parameters $p\in(0,1)$ and $q>0$ assigns to each subgraph $(V,A \subseteq E)$ a probability proportional to $p^{|A|}(1-p)^{|E|-|A|} q^{c(A)}$, where $c(A)$ is the number of connected components in~$(V,A)$. When $q=1$ this model corresponds to the standard \textit{bond percolation model}. For integer $q \ge 2$ the random-cluster model is closely related to the classical ferromagnetic \textit{$q$-state Ising/Potts model}. When $q\rightarrow 0$, the set of weak limits that arise contains the uniform measures over the spanning trees, spanning forests and connected subgraphs of $G$.

In this thesis we investigate the dynamics of the random-cluster model. While dynamics for the Ising/Potts model have been widely studied, random-cluster dynamics have so far largely resisted analysis. We focus on two canonical cases: the case when $G$ is the complete graph on $n$ vertices, known as the \textit{mean-field model}, and the case when $G$ is the infinite 2-dimensional lattice graph $Z^2$. Mean-field models have historically proven to be a useful starting point in understanding dynamics on more general graphs. In statistical mechanics, however, probabilistic models are most frequently studied in the setting of infinite lattice graphs; understanding random-cluster dynamics in $Z^2$ is thus of foremost importance.

In the first part of this thesis we establish the mixing time of Chayes-Machta dynamics in the the mean-field case. For $q \in (1,2]$ we prove that the mixing time is $\Theta(\log n)$ for all $p \neq p_c(q)$, where $p = p_c(q)$ is the critical value corresponding to the emergence of a ``giant'' component. For $q > 2$, we identify a critical window $(p_s,p_S)$ of the parameter $p$ around $p_c(q)$ in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is $\Theta(\log n)$ when $p \not\in [p_s,p_S]$ and $\exp(\Omega(\sqrt{n}))$ when $p \in (p_s,p_S)$. We also show that the mixing time is $\Theta(n^{1/3})$ for $p=p_s$ and $\Theta(\log n)$ for $p=p_S$. In addition, we prove that the Glauber dynamics undergoes a similar exponential slowdown in $(p_s,p_S)$.

The second part of this thesis focuses on the analysis of the Glauber dynamics of the random-cluster model in the case where the underlying graph is an $n \times n$ box in the Cartesian lattice~$Z^2$. Our main result is a tight $\Theta(n^2\log n)$ bound for the mixing time at all values of the model parameter~$p$ except the critical point $p=p_c(q)$, and for all values of~$q\ge 1$.