Monday, September 28, 2015
Klaus 1116 West - 1:00 pm
(Refreshments will be served in Klaus 2222 at 2 pm)
Title:
Counting hypergraph matchings up to uniqueness threshold
Abstract:
We study the problem of approximately counting hypergraph matchings with an activity parameter $\lambda$ in hypergraphs of bounded maximum degree and bounded maximum size of hyperedges. This problem unifies two important statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings).
We show 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. And we show that when $\lambda<\lambda_c$ the model exhibits strong spatial mixing at an exponential rate and there is an FPTAS for the partition function of the model on all hypergraphs of maximum degree at most $k+1$ and maximum edge size at most $d+1$. Assuming NP$\neq$RP, there is no FPRAS for the partition function of the model when $\lambda > 2\lambda_c$ on above family of hypergraphs .
Towards closing this gap and obtaining a tight transition of approximability, we study the local weak convergence from an infinite sequence of random finite hypergraphs to the infinite uniform regular hypertree with specified symmetry, and prove a surprising result: the existence of such local convergence is fully characterized by the reversibility of the uniform random walk on the infinite hypertree projected on the symmetry classes. We also give explicit constructions sequence of random finite hypergraphs with proper local convergence property when the reversibility condition is satisfied.
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