406 research outputs found
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
We establish the existence of free energy limits for several combinatorial
models on Erd\"{o}s-R\'{e}nyi graph and
random -regular graph . For a variety of models, including
independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy
both at a positive and zero temperature, appropriately rescaled, converges to a
limit as the size of the underlying graph diverges to infinity. In the zero
temperature case, this is interpreted as the existence of the scaling limit for
the corresponding combinatorial optimization problem. For example, as a special
case we prove that the size of a largest independent set in these graphs,
normalized by the number of nodes converges to a limit w.h.p. This resolves an
open problem which was proposed by Aldous (Some open problems) as one of his
six favorite open problems. It was also mentioned as an open problem in several
other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999
(Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob\'{a}s and Riordan
[Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin.
Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on
Discrete Structures (2004) 1-72 Springer].Comment: Published in at http://dx.doi.org/10.1214/12-AOP816 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Decay of Correlations for the Hardcore Model on the -regular Random Graph
A key insight from statistical physics about spin systems on random graphs is
the central role played by Gibbs measures on trees. We determine the local weak
limit of the hardcore model on random regular graphs asymptotically until just
below its condensation threshold, showing that it converges in probability
locally in a strong sense to the free boundary condition Gibbs measure on the
tree. As a consequence we show that the reconstruction threshold on the random
graph, indicative of the onset of point to set spatial correlations, is equal
to the reconstruction threshold on the -regular tree for which we determine
precise asymptotics. We expect that our methods will generalize to a wide range
of spin systems for which the second moment method holds.Comment: 39 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1004.353
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