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.
We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. ( 2020) and shows O(n log n) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(n log n) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree ∆ when λ < λ c (∆) where λ c (∆) is the critical point for the uniqueness/non-uniqueness phase transition on the ∆-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n log n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(n log n) mixing for q-colorings of triangle-free graphs of maximum degree ∆ when the number of colors satisfies q > α∆ where α ≈ 1.763, and O(m log n) mixing for generating random matchings of any graph with bounded degree and m edges.Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau (2020) to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan et al. (2019).
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