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Fluctuation results for multi-species Sherrington-Kirkpatrick model in the replica symmetric regime. (English) Zbl 1484.82064

Summary: We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in [D. Panchenko, Ann. Probab. 43, No. 6, 3494–3513 (2015; Zbl 1338.60237)], where the author proved the Parisi formula under positive-definite assumption on the disorder covariance matrix \(\Delta^2\). First, we prove exponential overlap concentration at high temperature for both indefinite and 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 positive-definite and emphindefinite \(\Delta^2\), 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 positive-definite and indefinite cases are different. Finally, in the case of 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 [E. Bates et al., J. Stat. Phys. 174, No. 2, 333–350 (2019; Zbl 1409.60138)], from 2-species to general species.

MSC:

82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60F05 Central limit and other weak theorems
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References:

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