Mixing Times of Critical 2D Potts Models
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The Potts model is a generalization of the Ising model to $q\geq 3$ states; on $\mathbb Z^d$ it is an extensively studied model of statistical mechanics, known to exhibit a rich phase transition for $d=2$ at some $\beta_c(q)$. Specifically, the Gibbs measure on $\mathbb Z^2$ exhibits a sharp transition between a disordered regime when $\beta<\beta_c(q)$ and an ordered regime when $\beta>\beta_c(q)$. At $\beta=\beta_c(q)$, when $q\leq 4$, the Potts model has a continuous phase transition and its scaling limit is believed to be conformally invariant; when $q>4$, the phase transition is discontinuous and the ordered and disordered phases coexist. I will discuss recent progress, joint with E. Lubetzky, in analyzing the time to equilibrium (mixing time) of natural Markov chains (e.g., heat-bath/Metropolis) for the 2D Potts model, where the mixing time on an $n\times n$ torus should transition from $O(\log n)$ at high temperatures to exponential in $n$ at low temperatures, via a critical slowdown at $\beta=\beta_c$ of $n^z$ when $q\leq 4$ and exponential in $n$ when $q>4$.
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