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 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 J. Stat. Phys. 174 (2019), no. 2, 333–350, from 2-species to general species.