Spin Systems on Bethe Lattices

Abstract

In an extremely influential paper Mézard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random d -regular graph [Eur. Phys. J. B 20 (2001) 217-233]. Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases. MSC: 05C80The following theorem establishes these conjectures rigorously. We say that G enjoys a property with high probability ('w.h.p.') if the probability that the property holds tends to one as n → ∞.Theorem 1.1. For any d ≥ 3, β > 0 the following is true. Let L = L(n) → ∞ be any integer sequence that tends to infinity. Then there exists a decomposition S 0 = S 0 (G), S 1 = S 1 (G), . . . , S ℓ = S ℓ (G), ℓ = ℓ(G) ≤ L, of the phase space {±1} n into non-empty sets such that µ G (S 0 ) = o(1) and such that with high probability (1.3)-(1.5) are satisfied for h = 1, . . . , ℓ.

The cut metric for probability distributions

“…Namely, while the graphon construction of the regular partition depends on delicately tracking a potential function, the pinning operation merely involves a purely mechanical reweighting of the probability distribution. The 'obliviousness' of the discrete pinning operation was vitally used in work on spin glass models on random graphs and on inference problems [8,9,10,11]. We show that a similarly oblivious procedure carries over naturally to the space of limit objects.…”

Free energy of a diluted spin glass model with quadratic Hamiltonian

“…In particular, within the Parisi framework of replica symmetry breaking theory, the work of Mézard and Parisi [13] proposed an ansatz to understand a spin glass model in the Bethe lattice. Later several attempts to justify the ansatz of [13] have been conducted in the setting of the diluted p-spin model and the diluted random K-SAT model, see [6,7,17,19,20,21]. Studies of high temperature behavior of diluted models have also been performed in the SK model [12], the V -statistics model [28], the random K-SAT model [15,18,25], and the Viana-Bray model [10].…”

The sparse parity matrix

“…This tells us that the proportion of frozen roots is φ d ( f (A) + o( 1)), provided that the newly added constraints do not dramatically shift the overall number of frozen variables due to long-range effects. To rule this out we use a delicate argument drawing on ideas from the study of random factor graph models and involving replica symmetry and the cut metric for discrete probability distributions from [5,14,17,18,19].…”

“…We establish this robustness by way of a pinning argument, in which unary checks are added that freeze certain previously unfrozen variables, and we analyse the effect that this has on the kernel. The thimblerig argument is an extension of arguments used in the study of random factor graph models [18,19,39], where the pinning operation also plays a crucial role [16,17].…”