Fluctuation of the Free Energy of Sherrington–Kirkpatrick Model with Curie–Weiss Interaction: The Paramagnetic Regime

Banerjee, Debapratim

doi:10.1007/s10955-019-02428-8

Cited by 7 publications

(9 citation statements)

References 23 publications

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“…This is strikingly similar to the interaction terms in the famous Sherrington-Kirkpatrick model. In the SK model, an expansion of the partition function can be expressed as a sum of products of weights around cycles as was shown in [Ban20]. While the partition function for the symmetric perceptron does not have an analogous expansion, we can still write Y as a similar sum over weighted cycles.…”

Section: Small Graph Conditioning For Dense Modelsmentioning

confidence: 99%

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

Abbé¹,

Li²,

Sly³

2021

Preprint

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We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15.We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mézard '89 in the asymmetric case.In a recent concurrent work of Perkins-Xu [PX21], the last two conjectures were also established by proving that the partition function concentrates on an exponential scale. This left open the contiguity conjecture and the lognormal limit characterization, which are established here. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

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Section: Small Graph Conditioning For Dense Modelsmentioning

confidence: 99%

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

Abbé¹,

Li²,

Sly³

2021

Preprint

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“…An important fact in (7) is that, in the regimes α 1/4, the contribution of large graphs to the partition function decays exponentially fast in |Γ|. This makes it possible to study the asymptotics of (7) combinatorically.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In contrast, in the super-critical regime, very large graphs are important and counting the cycles and paths becomes intractable. Restriced to the finite graph case in (7), a further reduction gives…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Compared to [2] and other works [1,7,8] built on it, the path cluster is a new structure that precisely suggests how the external field changes the behavior of the system from a combinatorial perspective. As we will prove later in Section 5, the first sum and the other sums for finitely many values of m, k in (8) are asymptotically independent Gaussian after appropriate centering; in particular…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Mean Field Spin Glass Models under Weak External Field

Dey¹,

Wu²

2021

Preprint

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We study the fluctuation and limiting distribution of free energy in mean-field spin glass models with Ising spins under weak external fields. We prove that at high temperature, there are three sub-regimes concerning the strength of external field h ≈ ρN −α with ρ, α ∈ (0, ∞). In the super-critical regime α < 1/4, the variance of the log-partition function is ≈ N 1−4α . In the critical regime α = 1/4, the fluctuation is of constant order but depends on ρ. Whereas, in the sub-critical regime α > 1/4, the variance is Θ(1) and does not depend on ρ. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One utilizes quadratic coupling and Guerra's interpolation scheme for Gaussian disorder, extending to many other spin glass models. However, this approach can prove the CLT only at very high temperatures. The other one is a cluster-based approach for general symmetric disorders, first used in the seminal work of Aizenman, Lebowitz, and Ruelle (Comm. Math. Phys. 112 (1987), no. 1, 3-20) for the zero external field case. It was believed that this approach does not work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the Sherrington-Kirkpatrick (SK) model when α 1/4. We further address the generality of this cluster-based approach. Specifically, we give similar results for the multi-species SK model and diluted SK model. Contents 1. Introduction and Main Results 2 2. Direct Proof in the Sub-Critical Case for Gaussian Disorder 11 3. Characteristic Approach for Gaussian Disorder 13 4. Cluster Based Approach for General Disorder 24 5. Multivariate CLT using Stein's Method 29 6. Computations for Stein's Method 34 7. Extension of Cluster Expansion to Other Models 40 8. Further Questions 44 References 46

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“…We now give a high-level sketch of the proof of the estimate Theorem 1.2 of log Z N in terms of cycle counts. The method has previously been used by the first author to study fluctuations in a stochastic block model [Ban+18], in a hypothesis testing problem for spiked random matrices [BM18] and in the SK model with Curie-Weiss interaction [Ban20].…”

Section: Previous Workmentioning

confidence: 99%

Fluctuations of the free energy of the mixed $p$-spin mean field spin glass model

Banerjee¹,

Belius²

2021

Preprint

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Fluctuation of the Free Energy of Sherrington–Kirkpatrick Model with Curie–Weiss Interaction: The Paramagnetic Regime

Cited by 7 publications

References 23 publications

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

Mean Field Spin Glass Models under Weak External Field

Fluctuations of the free energy of the mixed $p$-spin mean field spin glass model

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