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Spatial mixing and approximation algorithms for graphs with bounded connective constant | | Alistair Sinclair
; Piyush Srivastava
; Yitong Yin
; | | Date: |
8 Aug 2013 | | Abstract: | The hard core model in statistical physics is a probability distribution on
independent sets in a graph in which the weight of any independent set I is
proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show
that there is an intimate connection between the connective constant of a graph
and the phenomenon of strong spatial mixing (decay of correlations) for the
hard core model; specifically, we prove that the hard core model with vertex
activity lambda < lambda_c(Delta + 1) exhibits strong spatial mixing on any
graph of connective constant Delta, irrespective of its maximum degree, and
hence derive an FPTAS for the partition function of the hard core model on such
graphs. Here lambda_c(d) := d^d/(d-1)^(d+1) is the critical activity for the
uniqueness of the Gibbs measure of the hard core model on the infinite d-ary
tree. As an application, we show that the partition function can be efficiently
approximated with high probability on graphs drawn from the random graph model
G(n,d/n) for all lambda < e/d, even though the maximum degree of such graphs is
unbounded with high probability.
We also improve upon Weitz’s bounds for strong spatial mixing on bounded
degree graphs (Weitz, 2006) by providing a computationally simple method which
uses known estimates of the connective constant of a lattice to obtain bounds
on the vertex activities lambda for which the hard core model on the lattice
exhibits strong spatial mixing. Using this framework, we improve upon these
bounds for several lattices including the Cartesian lattice in dimensions 3 and
higher.
Our techniques also allow us to relate the threshold for the uniqueness of
the Gibbs measure on a general tree to its branching factor (Lyons, 1989). | | Source: | arXiv, 1308.1762 | | Services: | Forum | Review | PDF | Favorites | |
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