The SK Model Is Infinite Step Replica Symmetry Breaking at Zero Temperature

Auffinger, Antonio; Chen, Wei Kuo; Zeng, Qiang

doi:10.1002/cpa.21886

Cited by 23 publications

(20 citation statements)

References 53 publications

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“…For more on the relationship between replica overlaps and the Parisi minimizer, we refer the reader to [5][6][7], or to [33,45,46,36] for extended treatment of the subject.…”

Section: Introductionmentioning

confidence: 99%

Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

2019

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In the Sherrington-Kirkpatrick (SK) and related mixed p-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first nonasymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.2010 Mathematics Subject Classification. 60K35, 82B26, 82B44.

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“…For more on the relationship between replica overlaps and the Parisi minimizer, we refer the reader to [5][6][7], or to [33,45,46,36] for extended treatment of the subject.…”

Section: Introductionmentioning

confidence: 99%

Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

2019

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“…In particular, this conjecture is supported by high precision numerical solutions of the variational problem for P β [CR02, OSS07,SO08]. Rigorous evidence was recently obtained in [ACZ17]. Addressing this conjecture goes beyond the scope of the present paper.…”

Section: Introduction and Main Resultsmentioning

confidence: 51%

Optimization of the Sherrington-Kirkpatrick Hamiltonian

Montanari

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Let A ∈ R n×n be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing σ, Aσ over binary vectors σ ∈ {+1, −1} n . In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any ε > 0, outputs σ * ∈ {−1, +1} n such that σ * , Aσ * is at least (1 − ε) of the optimum value, with probability converging to one as n → ∞. The algorithm's time complexity is C(ε) n 2 . It is a message-passing algorithm, but the specific structure of its update rules is new.As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations.

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“…We prove eorem 1.1 by direct analysis of the bound in a speci c region of (y 1 , y 2 , Q). is seems to bear some resemblance to approaches of [CEJ + 18, ACZ17], although only at a high level. Our explicit choice of perturbation is based on the "bug proliferation" mechanism proposed by physicists [MR03,KPW04], which we detail in the introductory section below.…”

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confidence: 64%

“…e solution of the mean-eld variational formula is conjectured to be "full replica symmetry breaking" ( ), e.g. by analogy with the zero-temperature Sherrington-Kirkpatrick model [ACZ17]. A stronger version of has been shown in several recent works (in mean-eld se ings) to have algorithmic implications [AM18,Sub18,Mon18].…”

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confidence: 99%

Breaking of 1RSB in Random Regular MAX-NAE-SAT

Bartha

Sun

Zhang

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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A. For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satis ability transition occurs. For random k-and related models it happens at clause density α α sat 2 k . Just below the threshold, further results suggest that the solution space has a " " structure of a large bounded number of near-orthogonal clusters inside {0, 1} N .In the unsatis able regime α > α sat , it is natural to consider the problem of max-satis ability: violating the least number of constraints.is is a combinatorial optimization problem on the random energy landscape de ned by the problem instance. For a simpli ed variant, the strong refutation problem, there is strong evidence that an algorithmic transition occurs around α N k/2−1 . For α bounded in N, a very precise estimate of the max-sat value was obtained by Achlioptas, Naor, and Peres (2007), but it is not sharp enough to indicate the nature of the energy landscape. Later work (Sen, 2016; Panchenko, 2016) shows that for α very large (roughly, Ω(64 k )) the max-sat value approaches the mean-eld (complete graph) limit: this is conjectured to have an " " structure where near-optimal con gurations form clusters within clusters, in an ultrametric hierarchy of in nite depth inside {0, 1} N . A stronger form of was shown in several recent works to have algorithmic implications (again, in complete graphs). Consequently we nd it of interest to understand how the model transitions from near α sat , to (conjecturally) for large α. In this paper we show that in the random regular kmodel, the description breaks down already above α 4 k /k 3 . is is proved by an explicit perturbation in the parameter space. e choice of perturbation is inspired by the "bug proliferation" mechanism proposed by physicists (Montanari and Ricci-Tersenghi, 2003;Krzakala, Pagnani, and Weigt, 2004), corresponding roughly to a percolation-like threshold for a subgraph of dependent variables.

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The SK Model Is Infinite Step Replica Symmetry Breaking at Zero Temperature

Cited by 23 publications

References 53 publications

Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

Optimization of the Sherrington-Kirkpatrick Hamiltonian

Breaking of 1RSB in Random Regular MAX-NAE-SAT

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