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

Abstract

We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erd¨os-R´enyi 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 [Wor99], as well as [Ald],[BR],[JT08] and [AS03]). 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 Erd¨os-R´enyi 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 Erd¨os-R´enyi 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 NAEK- SAT. For example, let p(c, q,N) and p(r, q,N) denote, respectively, the probability that random graphs G(N, c/N) and G(N, r) are properly q-colorable. We prove the existence of limits of N[superscript −1] log p(c, q,N) and N[superscript 1] log p(r, q,N), as N -> infinity.