# Ising model - phase transition vs rapid mixing

Consider a graph $G=(V,E)$ and Ising model on that graph, i.e. configuration space is $\Omega=${$-1,+1$}$^V$ and energy of a configuration $s \in \Omega$ is given by: $H(s) = -\beta \sum_{u \sim v}s(u)s(v)$

Consider:

a) the phase transition between the ordered and disordered phase

b) transition in a Markov chain simulating the dynamics (e.g. Glauber/Metropolis dynamics), from rapid mixing ($N\log N$) to exponentially slow mixing

It is generally "known" that often phase transition as in a) is accompanied by phase transition in b) ("critical slowdown" in physics parlance). Is there any formal result capturing this knowledge (e.g. theorem of the form: physical phase transitions is equivalent to critical slowing down of the appropriate Markov chain)? Or, at least, a nonrigorous argument going beyond "this seems to hold for all systems that physicists are interested in" (proving tight results about critical points is usually difficult, so a heuristic would be OK).

• See recent papers of Lubetzky and Sly (and coauthors). – Steve Huntsman Dec 31 '12 at 2:16

We showed how in the Curie-Weiss model above critical temperature, a Metropolis chain has to make steps at rate $N$ (the number of spins) in order for the sample paths of the (suitably scaled) magnetization to converge to those of a Langevin diffusion with the correct limiting invariant distribution (a Gaussian). At the critical temperature the steps have to occur at the rate $N^{3/2}$, to obtain a Langevin diffusion for the invariant distribution proportional to $\exp(-x^4/12)$ in the proper scaling. The scaling in time is perhaps not exponential at the critical temperature because this is just about the magnetization, which does not describe the full state of the system.