Fluctuation results for Multi-species Sherrington-Kirkpatrick model in the replica symmetric regime

Abstract

We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann.~Probab.~43~(2015), no.~6, 3494--3513, where the author proved the Parisi formula under \emph{positive-definite} assumption on the disorder covariance matrix ${\Delta }^2$. First, we prove exponential overlap concentration at high temperature for both \indf~and \emph{positive-definite} ${\Delta }^2$ MSK model. We also prove a central limit theorem for the free energy using overlap concentration. Furthermore, in the zero external field case, we use a quadratic coupling argument to prove overlap concentration up to ${\beta }_c$, which is expected to be the critical inverse temperature. The argument holds for both \emph{positive-definite} and emph{indefinite} ${\Delta }^2$, and ${\beta }_c$ has the same expression in two different cases. Second, we develop a species-wise cavity approach to study the overlap fluctuation, and the asymptotic variance-covariance matrix of overlap is obtained as the solution to a matrix-valued linear system. The asymptotic variance also suggests the de Almeida--Thouless (AT) line condition from the Replica Symmetry (RS) side. Our species-wise cavity approach does not require the positive-definiteness of ${\Delta }^2$. However, it seems that the AT line conditions in \emph{positive-definite} and \emph{indefinite} cases are different. Finally, in the case of \emph{positive-definite} ${\Delta }^2$, we prove that above the AT line, the MSK model is in Replica Symmetry Breaking phase under some natural assumption. This generalizes the results of J.~Stat.~Phys.~174 (2019), no.~2, 333--350, from 2-species to general species.

35 pages, 1 figure. Final version. To appear in J.~Stat.~Phys

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Cited by in corpus (3)

• Free energy in multi-species mixed $p$-spin spherical models
• Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses
• Mean Field Spin Glass Models under Weak External Field