Abstract
Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous `cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007]. In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the $k$-XORSAT model and the diluted $k$-spin model for even $k$. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].
Highlights
Models based on random graphs have come to play a role in combinatorics, probability, statistics and computer science that can hardly be overstated
The random k-SAT model is of fundamental interest in computer science [4], the stochastic block model has gained prominence in statistics [1, 24, 36], low-density parity check codes have become a
Very similar models have been studied in statistical physics as models of disordered systems [31] and over the past 20 years physicists developed an analytic but non-rigorous technique for the study of such models called the ‘cavity method’
Summary
Models based on random graphs have come to play a role in combinatorics, probability, statistics and computer science that can hardly be overstated. Very similar models have been studied in statistical physics as models of disordered systems [31] and over the past 20 years physicists developed an analytic but non-rigorous technique for the study of such models called the ‘cavity method’ This nonrigorous approach has inspired numerous “predictions” with an impact on an astounding variety of problems (e.g., [15, 31, 33, 42]). While the cavity method can be applied almost mechanically to a wide variety of problems, most rigorous arguments still hinge on model-specific deliberations, a state of affairs that begs the questions of whether we can rigorise the physics calculations wholesale. This is the thrust of the present paper.
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