Charting the replica symmetric phase

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. in Proc Natl Acad Sci 104:10318–10323, 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. in Phys Rev E 84:066106, 2011)

The satisfiability threshold for random linear equations

Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear system $A x=y$ has a solution with high probability and analyse the geometry of the set of solutions. In the special case $q=2$, known as the random $k$-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof technique was subsequently extended to the cases $q=3,4$ [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to $q>3$. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof

Strong replica symmetry in high-dimensional optimal Bayesian inference

We consider generic optimal Bayesian inference, namely, models of signal reconstruction where the posterior distribution and all hyperparameters are known. Under a standard assumption on the concentration of the free energy, we show how replica symmetry in the strong sense of concentration of all multioverlaps can be established as a consequence of the Franz-de Sanctis identities; the identities themselves in the current setting are obtained via a novel perturbation coming from exponentially distributed "side-observations" of the signal. Concentration of multioverlaps means that asymptotically the posterior distribution has a particularly simple structure encoded by a random probability measure (or, in the case of binary signal, a non-random probability measure). We believe that such strong control of the model should be key in the study of inference problems with underlying sparse graphical structure (error correcting codes, block models, etc) and, in particular, in the rigorous derivation of replica symmetric formulas for the free energy and mutual information in this context

The ising antiferromagnet and max cut on random regular graphs

The Ising antiferromagnet is an important statistical physics model with close connections to the MAX CUT problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the MAX CUT of random regular graphs predicted by Zdeborová and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound

Belief Propagation on the random kk-SAT model

Corroborating a prediction from statistical physics, we prove that the Belief Propagation message passing algorithm approximates the partition function of the random $k$-SAT model well for all clause/variable densities and all inverse temperatures for which a modest absence of long-range correlations condition is satisfied. This condition is known as "replica symmetry" in physics language. From this result we deduce that a replica symmetry breaking phase transition occurs in the random $k$-SAT model at low temperature for clause/variable densities below but close to the satisfiability threshold