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.

中文翻译：

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通过马尔可夫链在低温下实现快速算法†

在自旋系统中进行近似计数和采样的有效算法通常适用于所谓的高温状态，其中相邻自旋之间的相互作用是“弱”的。取而代之的是，Jenssen，Keevash和Perkins的最新工作在低温状态下使用聚合物模型和聚类展开在有界度（二分）展开图上生成了多项式时间算法。为了加快这些算法的速度（因此，运行时的指数不取决于度数的界线），我们提出了一种用于聚合物模型的马尔可夫链，并表明它在聚合物重量的指数衰减下迅速混合。这样会产生有限度展开图上的低温铁磁Potts模型的时间采样算法。将我们的硬核模型和Potts模型的结果与Markov链比较工具相结合，我们获得了限于状态空间适当部分的Glauber动力学的多项式混合时间。