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

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 app...

• Type: Conference
• Year: 2010
• Where: STOC
• Authors: Mohsen Bayati, David Gamarnik, Prasad Tetali
• Keywords: Algorithms, Erd¨os-r´enyi Graph, Free Energy, STOC 2010, Zero Temperature