Uniqueness and Rapid Mixing in the Bipartite Hardcore Model (extended abstract)

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 is originated in statistical physics for modeling equilibrium of lattice gas. Combinatorially, it can also be seen as a weighted enumeration of independent sets. Counting the number of independent sets in a bipartite graph (#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. Interestingly, we are able to show that a regime that was considered "low-temperature" is actually well within the uniqueness (high-temperature) regime. 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 (STOC 2015), and it makes use of a connection between correlation decay and spectral independence of Markov chains. Along the way, we also build an explicit connection between the very recent developments of negative-fields stochastic localization schemes and field dynamics. 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. Remarkably, this is a model where both the total influence and the spectral radius of the adjacency matrix can be unbounded, yet we are able to prove mixing time bounds through the framework of spectral independence.