Critical slowdown in terms of Markov process? | PhysicsOverflow
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

157 submissions , 130 unreviewed
4,116 questions , 1,513 unanswered
4,971 answers , 21,204 comments
1,470 users with positive rep
571 active unimported users
More ...

  Critical slowdown in terms of Markov process?

+ 4 like - 0 dislike

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? 

asked Sep 12, 2016 in Computational Physics by Jia Yiyang (2,635 points) [ revision history ]

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.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights