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

TL;DR

This paper proves the existence of free energy limits and scaling limits for various combinatorial models on Erdős-Rényi and random regular graphs, resolving several longstanding open problems in the field.

Contribution

It introduces a combinatorial approach to establish the convergence of free energy and scaling limits for models like independent sets, MAX-CUT, coloring, and K-SAT on random graphs.

Findings

Free energy limits exist for multiple models on Erdős-Rényi and regular graphs.

Normalized size of the largest independent set converges to a limit with high probability.

Resolves an open problem proposed by Aldous regarding the scaling limit of independent sets.

Abstract

We establish the existence of free energy limits for several combinatorial models on Erd\"{o}s-R\'{e}nyi graph $\mathbb {G}(N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb {G}(N,r)$. 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. 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].