Abstract
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 ‐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.
Highlights
The hard-core model from statistical physics is defined on the set of independent sets of a graph G, where the independent sets are weighted by a fugacity λ > 0
We apply our results for subset polymer models to two specific examples: the ferromagnetic Potts model and the hard-core model on expander graphs
A very natural idea to sample at low temperatures is to use a single-spin update Markov chain like the Glauber dynamics, but to start in one of the ground states of the model chosen at random
Summary
The hard-core model from statistical physics is defined on the set of independent sets of a graph G, where the independent sets are weighted by a fugacity λ > 0. In general there are no known efficient algorithms at low temperatures (high fugacities), but recently efficient algorithms have been developed for some special classes of graphs including subsets of Zd [18], random regular bipartite graphs, and bipartite expander graphs in general [21, 25] What these bipartite graphs have in common is that for large enough λ, typical independent sets drawn from μG,λ align closely with one side or the other of the bipartition (the two ground states). Sampling algorithms for low temperature models on expander graphs in cases where only nO(log Δ)-time algorithms were previously known
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