  We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter $\lambda$, each matching $M$ is assigned a weight $\lambda^{|M|}$.
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
  For this model, the critical activity $\lambda_c= \frac{d^d}{k (d-1)^{d+1}}$
is the threshold for the uniqueness of Gibbs measures on the infinite
$(d+1)$-uniform $(k+1)$-regular hypertree. Consider hypergraphs of maximum
degree at most $k+1$ and maximum size of hyperedges at most $d+1$. We show that
when $\lambda < \lambda_c$, there is an FPTAS for computing the partition
function; and when $\lambda = \lambda_c$, there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When $\lambda >
2\lambda_c$, there is no PRAS for the partition function or the log-partition
function unless NP$=$RP.
  Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets.
