On the energy landscape of the mixed even p-spin model

Chen, Wei Kuo; Handschy, Madeline; Lerman, Gilad

doi:10.1007/s00440-017-0773-1

Cited by 34 publications

(66 citation statements)

References 46 publications

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“…We remark that the same formula for the discrepancy of the entropy of a pure state (defined in the next section) also holds at low temperature, which can be seen from Lemma 6 below. Theorem 3 combined with Lemma 4.4 in [7] implies that the Wasserstein distance with the Hamming cost function d(σ 1 , σ 2 ) = N −1 ∑ i≤N I(σ 1 i = σ 2 i ) between G N and the product measure with the same marginals does not go to zero even when (15) holds. This gives a negative answer to the Conjecture 1.4.18 in [39].…”

Section: High Temperature Casementioning

confidence: 99%

“…Moreover, under the condition α P ({q P }) > 0, the measure G is supported by countably many points (pure states) each carrying positive random weight (see Lemma 2.7 in [27]). It was explained in Jagannath [21] how the ultrametric structure of the asymptotic Gibbs measure G can be used to define approximate pure states for finite N, which are clusters of spin configurations on {−1, +1} N that satisfy various natural properties, most importantly, the analogue of (15). Namely, with respect to the conditional Gibbs measure on a given cluster, the overlap R 1,2 concentrates near q P .…”

Section: Low Temperature Casementioning

confidence: 99%

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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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Section: High Temperature Casementioning

confidence: 99%

Section: Low Temperature Casementioning

confidence: 99%

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

Chen

Panchenko

2018

Commun. Math. Phys.

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“…[Remark: In fact, the case of K-spin model was established in Theorem 2 in [11] and stated for t ∈ (0, 1) in both [11,12]. However, the case of t = 0 is implicitly included in the proof of Theorem 2 in [11] and is in fact the easiest case.…”

Section: )mentioning

confidence: 99%

“…with the boundary condition Φ γ (1, x) = |x|. 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.…”

Section: The Parisi Formula and Guerra-talagrand Boundmentioning

confidence: 99%

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

Auffinger

Chen

Zeng

2020

Comm Pure Appl Math

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We prove that the Parisi measure of the mixed p-spin model at zero temperature has infinitely many points in its support. This establishes Parisi's prediction that the functional order parameter of the Sherrington-Kirkpatrick model is not a step function at zero temperature. As a consequence, we show that the number of levels of broken replica symmetry in the Parisi formula of the free energy diverges as the temperature goes to zero.

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On the energy landscape of the mixed even p-spin model

Cited by 34 publications

References 46 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

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

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