TAP approach for multi-species spherical spin glasses I: general theory

TL;DR

This paper extends the TAP approach to multi-species spherical spin glasses, providing a generalized representation of free energy and insights into the Onsager correction for maximal overlaps.

Contribution

It develops a generalized TAP framework for multi-species spherical mixed p-spin models, including a representation of free energy at various overlaps.

Findings

Proves a generalized TAP representation for free energy at any multi-samplable overlap.

Shows the TAP correction equals the Onsager reaction term at maximal overlaps.

Lays groundwork for computing free energy in multi-species pure p-spin models in a companion paper.

Abstract

We develop a generalized TAP approach for the multi-species version of the spherical mixed $p$-spin models. In particular, we prove a generalized TAP representation for the free energy at any overlap vector which is multi-samplable in an appropriate sense. Moreover, we show that if a multi-samplable overlap is maximal, then the TAP correction is equal to an analogue of the well-known Onsager reaction term. Finally, in a companion paper we use the results from the current paper to compute the free energy at any temperature for all multi-species pure $p$-spin models, assuming the free energy converges.