NDLI: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

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Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

Content Provider	ACM Digital Library
Author	Bayati, Mohsen Gamarnik, David Tetali, Prasad
Abstract	We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erdos-Renyi (ER) graph G(N,c/N) and random r-regular graph G(N,r). 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., thus resolving an open problem, (see Conjecture 2.20 in Wormald's Random Graph Models, or Aldous's list of open problems. Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli. Among other applications, this method was used to prove the existence of free energy limits for Viana-Bray and K-SAT models on ER graphs. The case of zero temperature was treated by taking limits of positive temperature models. We provide instead a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of ER graph G(N,c/N) and random regular graph G(N,r). In addition we establish the large deviations principle for the satisfiability property for constraint satisfaction problems such as Coloring, K-SAT and NAE-K-SAT.
Starting Page	105
Ending Page	114
Page Count	10
File Format	PDF
ISBN	9781450300506
DOI	10.1145/1806689.1806706
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2010-06-05
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Partition function Random graphs Scaling limit Constraint satisfaction problems
Content Type	Text
Resource Type	Article

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