Fluctuation Results for Multi-species Sherrington-Kirkpatrick Model in the Replica Symmetric Regime

Dey, Partha S.; Qiang, Wu

doi:10.1007/s10955-021-02835-w

Cited by 9 publications

(7 citation statements)

References 25 publications

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“…. , m. In the zero external field case the critical β is given by β c := Λ 1/2 ∆ 2 Λ 1/2 −1/2 (see [20]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this proof, the general framework is based on the smart path interpolation and characteristic approach. The idea was also used in [20] for proving the central limit theorem of free energy in the multi-species SK model. Gaussianity of the disorder is essential to the proof as Gaussian integration by parts was utilized multiple times.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…As in the previous section, we can assume that it has a symmetric distribution with E J ij = 0, E J 2 ij = 1 and finite (4 + ε)-th moment for some ε > 0. The fluctuation results were obtained in [20] for h > 0 and in [32] for h = 0. For h = h N , our main goal is to illustrate how the cluster based approach can be easily extended to the current setting, which gives the fluctuation results under weak external fields.…”

Section: The Hamiltonian Of the Bipartite Model Is Given Bymentioning

confidence: 97%

“…For various aspects of this model, we refer readers to [10-12, 20, 34] and references therein. We further point out that the bipartite model can be realized as an instance of the MSK model, see [20].…”

Section: Extension Of Cluster Expansion To Other Modelsmentioning

confidence: 99%

“…By the discussions in Section 3.2, we need to prove the L 1 convergence in (20) for the subcritical and critical cases. The inequality (22) suggests that it is enough to prove the results in (23).…”

Section: Now We Prove Theorem 34mentioning

confidence: 99%

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Mean Field Spin Glass Models under Weak External Field

Dey¹,

Wu²

2021

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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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“…. , m. In the zero external field case the critical β is given by β c := Λ 1/2 ∆ 2 Λ 1/2 −1/2 (see [20]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%