  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.
