# Critical slowdown in terms of Markov process?

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In various local update schemes of Monte Carlo method (such as Metropolis algorithm), there's a general phenomenon known as critical slowdown, which is the slowdown of convergence to thermal equilibrium near critical temperature. This is intuitively easy to understand: a long range correlation needs to be established near critical point, while local updates only update the system in very short ranges at each step.

I wonder if there's a more mathematically precise way of proving this. In particular, since every Monte Carlo method is essentially a Markov process with equilibrium configuration having the largest eigenvalue (which is 1) for the Markov matrix, a slowdown has to mean there's a second eigenvalue extremely near 1, how do we prove this, say, for Metropolis algorithm?

Self-similarity up to arbitrarily large scales implies an accumulation of eigenvectors approaching 1 (if there is a relaxation mode at scale $k$ with frequency $\omega(k)$, then there is typically also a mode at scale $\alpha k$ with frequency $~\alpha^\gamma \omega(k)$). I'm not sure exactly why the frequencies $\omega(k)$ typically behave this way however---it could just be an artifact of using local Gaussian noise as a way to drive the kinds of SPDE's where this sort of behavior appears.

For common implementations of the metropolis algorithm, you make local random adjustments to spin degrees of freedom in bounded domains (often a single lattice site). In lattice models near criticality, the eigenvectors of the Markov matrix consist of linear combinations of field configurations that differ from each other over very large length scales, and hence typical instances are very far apart with respect to the natural metric of the most commonly used metropolis algorithm. Viewing the metropolis algorithm as a type of random walk, the relaxation time would be controlled (though not optimally) by the number of local rearrangements that are needed to change between two very different typical configurations of spins, which scales positively with the system size.

@Henry, thanks for the comment.

it could just be an artifact of using local Gaussian noise as a way to drive the kinds of SPDE's where this sort of behavior appears.

You may be right, I read a claim that the so called "worm algorithm" is in fact efficient near critical point, albeit being a local update scheme. But I need to read more on this.

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