Limits of discrete distributions and Gibbs measures on random graphs

Coja-Oghlan, Amin; Perkins, Will; Skubch, Kathrin

doi:10.1016/j.ejc.2017.06.012

Cited by 17 publications

(45 citation statements)

References 33 publications

(80 reference statements)

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“…This leads to a general notion of limits of probability measures on discrete cubes. The article also discusses the connection with the Aldous‐Hoover representation of exchangeable arrays, which has long been known to be related to the theory of graph limits, and Panchenko's notion of asymptotic Gibbs measures . A further recent application of the methods of the present paper to two special classes of random factor graph models can be found in .…”

Section: Introductionmentioning

confidence: 96%

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Harnessing the Bethe free energy

Bapst

Coja-Oghlan

2016

Random Struct Algorithms

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A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the k‐SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the “replica symmetric cavity method” yields the correct value of the partition function if the underlying model enjoys certain properties [Krzkala et al., PNAS (2007) 10318–10323]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a “regularity lemma” for probability measures on sets of the form Ωn for a finite Ω and a large n that may be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 694–741, 2016

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Section: Introductionmentioning

confidence: 96%

“…The material of Section 2 of the present paper has recently been investigated from the more analytic viewpoint of the theory of graph limits . This leads to a general notion of limits of probability measures on discrete cubes.…”

Section: Introductionmentioning

confidence: 99%

Harnessing the Bethe free energy

Bapst

Coja-Oghlan

2016

Random Struct Algorithms

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“…Technically the paper builds upon and continues two intertwined threads of prior work. First, we bring to bear a variant of the 'regularity method' from combinatorics that we developed recently [10,18,19] in order to establish the pure state decomposition and to vindicate the Belief Propagation equations. Second, we seize upon Panchenko's work on asymptotic Gibbs measures and the interpolation method, particularly in order to derive the variational formula for the free energy [52,53].…”

Section: Introductionmentioning

confidence: 99%

Spin Systems on Bethe Lattices

Coja-Oghlan

Perkins

2019

Commun. Math. Phys.

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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.

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“…This suggest that something non-trivial is happening for 7-circular coloring of random 3 regular graphs. And the upper bound established by [11] is non-trivial along the lines of the upper bound of [13] for coloring.…”

Section: A Context In Mathematicsmentioning

confidence: 99%

“…For a given reweighting parameter m the 1RSB equation (10) can be solved with the population dynamics technique. Solving it for every m yields the thermodynamic quantities as function of the reweighting parameter m. These are Φ s (m) which follows from (14), Σ(m) which is obtained from (13) and s(m) which can be computed with (5) and (6) -all in the limit β → ∞ where −βf = s. In order to assign the right weights to each pure state in the equilibrium configuration (1RSB estimate), the reweighting parameter m must be chosen accordingly. If Σ(m = 1) > 0 we are in the dynamic 1RSB phase and the 1RSB and RS solutions agree.…”

Section: Rsb In the Entropic Zero Temperature Limitmentioning

confidence: 99%

Circular coloring of random graphs: statistical physics investigation

2016

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Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.

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Limits of discrete distributions and Gibbs measures on random graphs

Cited by 17 publications

References 33 publications

Harnessing the Bethe free energy

Harnessing the Bethe free energy

Spin Systems on Bethe Lattices

Circular coloring of random graphs: statistical physics investigation

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