Parisi formula for the ground state energy in the mixed $p$-spin model

Auffinger, Antonio; Chen, Wei Kuo

doi:10.1214/16-aop1173

Cited by 48 publications

(59 citation statements)

References 29 publications

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“…Recall the Parisi functional P from (5). Also recall that the Parisi measure α P (ds) is the unique minimizer of P and q P is the largest point in the support of α P (ds).…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the TAP Free Energy in the Mixed p-Spin Models

Chen

Panchenko

2018

Commun. Math. Phys.

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In [41], Thouless, Anderson, and Palmer derived a representation for the free energy of the Sherrington-Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended configuration space of local magnetizations with an Onsager correction term. In the setting of mixed p-spin models with Ising spins, we prove that the free energy can indeed be written as the supremum of the TAP free energy over the space of local magnetizations whose Edwards-Anderson order parameter (self-overlap) is to the right of the support of the Parisi measure. Furthermore, for generic mixed p-spin models, we prove that the free energy is equal to the TAP free energy evaluated on the local magnetization of any pure state.

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“…Recall the Parisi functional P from (5). Also recall that the Parisi measure α P (ds) is the unique minimizer of P and q P is the largest point in the support of α P (ds).…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the TAP Free Energy in the Mixed p-Spin Models

Chen

Panchenko

2018

Commun. Math. Phys.

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“…The proof of Theorem 3 will utilize the zero temperature Parisi formula recently established by Auffinger and Chen in [2], and the Guerra-Talagrand replica symmetry breaking bound at zero temperature, which is an extension of the corresponding bound at positive temperature used earlier by Guerra and Talagrand [21,42,43,44] to study the mean field K-spin model. Similar techniques have recently been used in [8] to study the questions of dynamics at positive temperature.…”

Section: Overlap Gap Propertymentioning

confidence: 99%

“…The existence, uniqueness, and regularity properties of the solution Φ γ (s, x) were studied in [11,Appendix]. The Parisi formula for the maximum energy in [2] states ME := lim N→∞ ME N = inf γ∈U P(γ), (5.5) where the limit of ME N exists almost surely. Indeed, (5.5) was established in [2] for general mixed p-spin models by showing how the corresponding formulas at positive temperature, first proved for mixed even pspin models in [43] and for general mixed p-spin models in [34,35], are transformed in the zero-temperature limit.…”

Section: The Parisi Formula and Guerra-talagrand Boundmentioning

confidence: 99%

Suboptimality of local algorithms for a class of max-cut problems

Chen¹,

Gamarnik²,

Panchenko³

et al. 2019

Ann. Probab.

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We show that in random K-uniform hypergraphs of constant average degree, for even K ≥ 4, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain non-trivial interval -a phenomenon referred to as the overlap gap property -which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models, and showing the overlap gap property in the latter setting. This paper considers the problem of algorithmically finding nearly optimal spin configurations in the diluted K-spin model. We specifically focus on local algorithms defined as factors of i.i.d., the formal definition of which is provided in Section 2. The diluted K-spin model is also known as the max-cut problem for Kuniform Erdős-Rényi hypergraphs of constant average degree, and also as the random K-XORSAT model. The problem is only interesting for even K and we prove that, for even K ≥ 4, local algorithms fail to find the nearly optimal spin configurations (maximal cuts) once the average degree is large enough.The proof is based on finding a structural constraint for the overlap of any two nearly optimal spin configurations -the overlap gap property -that goes against certain properties of local algorithms. For K = 2, the overlap gap property is not expected to hold, which is why this case is excluded. The structural constraint is derived from recent results on the mean field K-spin spin glass models, in particular, the Parisi formula and the Guerra-Talagrand replica symmetry breaking bound at zero temperature. We begin with a discussion of the model and the notion of algorithms that we use.The K-spin model The set of ±1 spin configurations on N vertices will be denoted by

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“…It turns out that, for large α, optimal assignments are not much better than any fixed assignment and the leading term of the smallest ratio of unsatisfied clauses to the number of variables N is α/2 K . We will show that for optimal assignments the next order correction term for large α is of the form −c * √ α, where the constant c * = c * (N) is related to the expected maximum of the specific mixed p-spin spin glass model in (4) below. This will establish the Leuzzi-Parisi formula obtained in [16] by the non-rigorous replica method.…”

Section: Introductionmentioning

confidence: 97%

“…The good news is that, due to a recent breakthrough in (building upon the ideas in ), the zero temperature limit of the Parisi formula can be expressed in a form (conjectured by Guerra) quite similar to the classical Parisi formula at positive temperature, as follows. Let

normalU

be the family of all nonnegative nondecreasing step functions on

[0, 1]

with finitely many jumps.…”

Section: Introductionmentioning

confidence: 99%

On the K‐sat model with large number of clauses

Panchenko

2017

Random Struct Algorithms

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We show that in the K‐sat model with N variables and αN clauses, the expected ratio of the smallest number of unsatisfied clauses to the number of variables is α/2K−normalαc∗(N)/2K up to smaller order terms o(normalα) as α→∞ uniformly in N, where c∗(N) is the expected normalized maximum energy of some specific mixed p‐spin spin glass model. The formula for the limit of c∗(N) is well known in the theory of spin glasses.

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Parisi formula for the ground state energy in the mixed $p$-spin model

Abstract: We show that the thermodynamic limit of the ground state energy in the mixed pspin model can be identified as a variational problem. This gives a natural generalization of the Parisi formula at zero temperature.

Cited by 48 publications

References 29 publications

On the TAP Free Energy in the Mixed p-Spin Models

On the TAP Free Energy in the Mixed p-Spin Models

Suboptimality of local algorithms for a class of max-cut problems

On the K‐sat model with large number of clauses

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