Fast algorithms at low temperatures via Markov chains

Efficient algorithms for approximate counting and sampling in spin systems typically apply in the so‐called high‐temperature regime, where the interaction between neighboring spins is “weak.” Instead, recent work of Jenssen, Keevash, and Perkins yields polynomial‐time algorithms in the low‐temperatu...

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Bibliographic Details
Published in:Random structures & algorithms 2021-03, Vol.58 (2), p.294-321
Main Authors: Chen, Zongchen, Galanis, Andreas, Goldberg, Leslie A., Perkins, Will, Stewart, James, Vigoda, Eric
Format: Article
Language:English
Subjects:
Algorithms
approximate counting
expander graphs
Ferromagnetism
Graphs
hard‐core model
Low temperature
Markov analysis
Markov chains
Polymers
Polynomials
Potts model
Run time (computers)
Sampling
Temperature
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Summary:Efficient algorithms for approximate counting and sampling in spin systems typically apply in the so‐called high‐temperature regime, where the interaction between neighboring spins is “weak.” Instead, recent work of Jenssen, Keevash, and Perkins yields polynomial‐time algorithms in the low‐temperature regime on bounded‐degree (bipartite) expander graphs using polymer models and the cluster expansion. In order to speed up these algorithms (so the exponent in the run time does not depend on the degree bound) we present a Markov chain for polymer models and show that it is rapidly mixing under exponential decay of polymer weights. This yields, for example, an O(nlogn)‐time sampling algorithm for the low‐temperature ferromagnetic Potts model on bounded‐degree expander graphs. Combining our results for the hard‐core and Potts models with Markov chain comparison tools, we obtain polynomial mixing time for Glauber dynamics restricted to appropriate portions of the state space.
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20968