4,181 research outputs found
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
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
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , 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 , there is no PRAS for the partition function or the log-partition
function unless NPRP.
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
Birthday Inequalities, Repulsion, and Hard Spheres
We study a birthday inequality in random geometric graphs: the probability of
the empty graph is upper bounded by the product of the probabilities that each
edge is absent. We show the birthday inequality holds at low densities, but
does not hold in general. We give three different applications of the birthday
inequality in statistical physics and combinatorics: we prove lower bounds on
the free energy of the hard sphere model and upper bounds on the number of
independent sets and matchings of a given size in d-regular graphs.
The birthday inequality is implied by a repulsion inequality: the expected
volume of the union of spheres of radius r around n randomly placed centers
increases if we condition on the event that the centers are at pairwise
distance greater than r. Surprisingly we show that the repulsion inequality is
not true in general, and in particular that it fails in 24-dimensional
Euclidean space: conditioning on the pairwise repulsion of centers of
24-dimensional spheres can decrease the expected volume of their union
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