On the de Almeida--Thouless Transition Surface in the Multi-Species SK Model with Centered Gaussian External Field

Abstract

We study the phase transition of the Parisi formula for the free energy in the multi-species Sherrington--Kirkpatrick model with a centered Gaussian external field and a positive-semidefinite variance profile matrix. We show that in terms of the strength of the external field and the variance profile, the de Almeida--Thouless surface delineates the boundary between replica symmetric solutions and replica symmetry breaking solutions.

54 pages, 1 figure. Strengthened Proposition 3.1 to include external fields, revised Proposition 4.1 and the proof of the main theorem accordingly, and made minor revisions and updated acknowledgments