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Tight bounds on the coefficients of partition functions via stability. (arXiv:1704.07784v1 [math.CO])
来源于:arXiv
Partition functions arise in statistical physics and probability theory as
the normalizing constant of Gibbs measures and in combinatorics and graph
theory as graph polynomials. For instance the partition functions of the
hard-core model and monomer-dimer model are the independence and matching
polynomials respectively.
We show how stability results follow naturally from the recently developed
occupancy method for maximizing and minimizing physical observables over
classes of regular graphs, and then show these stability results can be used to
obtain tight extremal bounds on the individual coefficients of the
corresponding partition functions.
As applications, we prove new bounds on the number of independent sets and
matchings of a given size in regular graphs. For large enough graphs and almost
all sizes, the bounds are tight and confirm the Upper Matching Conjecture of
Friedland, Krop, and Markstr\"om and a conjecture of Kahn on independent sets
for a wide range of parameters. Additi 查看全文>>