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It is physically understood why the standard Metropolis-Hasting algorithm slows down near the critical temperature, since it doesn’t utilize the divergence of the correlation length. However, I’m surprised that I couldn’t find a mathematical proof on this topic considering how old it is. I’m especially surprised that cluster update algorithms (as a solution to critical slow-down) also do not seem to have a proof of convergence rate despite it being based on FK-percolation which has a plethora of literature on it.

Question: Does anyone know how to prove the existence of critical slow-down of the standard Metropolis Hasting algorithm? Or alternatively, prove that cluster update algorithms can bypass this problem?

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    $\begingroup$ As far as I know, this is not yet fully understood from a mathematical point of view (except maybe on trees or complete graphs?). However there are some interesting results, for instance the papers by Lubetzky, Sly and collaborators, including this one and this one (and presumably others; I haven't followed this line of research closely enough). $\endgroup$
    – Yvan Velenik
    Commented Apr 24 at 15:42
  • $\begingroup$ The references look interesting. Thx! $\endgroup$
    – Andrew Yuan
    Commented Apr 24 at 19:14

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