We design an efficient sampling algorithm to generate samples from the hardcore model on random regular bipartite graphs as long as $λ\lesssim \frac{1}{\sqrtΔ}$, where $Δ$ is the degree. Combined with recent work of Jenssen, Keevash and Perkins this implies an FPRAS for the partition function of the hardcore model on random regular bipartite graphs at any fugacity. Our algorithm is shown by analyzing two new Markov chains that work in complementary regimes. Our proof then proceeds by showing the corresponding simplicial complexes are top-link spectral expanders and appealing to the trickle-down theorem to prove fast mixing.

翻译：我们设计了一种高效抽样算法，用于从随机正则二分图的硬核模型中生成样本，只需满足$λ\lesssim \frac{1}{\sqrtΔ}$，其中$Δ$是度数。结合Jenssen、Keevash和Perkins近期的工作，这意味着在任意逸度下，对随机正则二分图的硬核模型配分函数存在一个FPRAS。该算法通过分析两个互补区间内工作的新马尔可夫链来证明。我们的证明随后通过展示对应的单纯复形是拓扑链谱扩展子，并借助渗流定理来证明快速混合。