We characterize the uniqueness condition in the hardcore model for bipartite graphs with degree bounds only on one side, and provide a nearly linear time sampling algorithm that works up to the uniqueness threshold. We show that the uniqueness threshold for bipartite graph has almost the same form of the tree uniqueness threshold for general graphs, except with degree bounds only on one side of the bipartition. The hardcore model from statistical physics can be seen as a weighted enumeration of independent sets. Its bipartite version (#BIS) is a central open problem in approximate counting. Compared to the same problem in a general graph, surprising tractable regime have been identified that are believed to be hard in general. This is made possible by two lines of algorithmic approach: the high-temperature algorithms starting from Liu and Lu (STOC 2015), and the low-temperature algorithms starting from Helmuth, Perkins, and Regts (STOC 2019). In this work, we study the limit of these algorithms in the high-temperature case. Our characterization of the uniqueness condition is obtained by proving decay of correlations for arguably the best possible regime, which involves locating fixpoints of multivariate iterative rational maps and showing their contraction. We also give a nearly linear time sampling algorithm based on simulating field dynamics only on one side of the bipartite graph that works up to the uniqueness threshold. Our algorithm is very different from the original high-temperature algorithm of Liu and Lu, and it makes use of a connection between correlation decay and spectral independence of Markov chains. Last but not the least, we are able to show that the standard Glauber dynamics on both side of the bipartite graph mixes in polynomial time up to the uniqueness.

翻译：我们刻画了仅在一侧有度约束的二分图硬核模型的唯一性条件，并提出了一个在唯一性阈值内有效的近线性时间采样算法。我们证明，二分图上的唯一性阈值与一般图上的树唯一性阈值形式几乎相同，唯一的区别是仅对二分图的一侧施加度约束。统计物理中的硬核模型可视为对独立集的加权枚举。其二分图版本（#BIS）是近似计数领域的核心开放问题。与一般图中的同类问题相比，研究者已识别出在二分图中出人意料的可解区域，而这些区域在一般图中被认为难以处理。这得益于两类算法途径：始于Liu和Lu（STOC 2015）的高温算法，以及始于Helmuth、Perkins和Regts（STOC 2019）的低温算法。本文研究了这些算法在高温情形下的极限。我们通过证明可能最佳范围内的相关性衰减来刻画唯一性条件，这涉及定位多元迭代有理映射的不动点并证明其收缩性。我们还提出了一种仅基于二分图一侧场动力学模拟的近线性时间采样算法，该算法在唯一性阈值内有效。与Liu和Lu最初的高温算法不同，我们的算法利用了相关性衰减与马尔可夫链谱独立性之间的联系。最后同样重要的是，我们能够证明在唯一性阈值内，二分图双侧的标准格劳伯动力学可在多项式时间内混合。