The Parisi formula for mixed $p$-spin models

TL;DR

This paper extends the proof of the Parisi formula for mixed p-spin models to include odd p interactions, completing the understanding of free energy in these complex spin glass systems.

Contribution

It provides the first proof of the Parisi formula for general mixed p-spin models, including odd p interactions, building on recent ultrametricity results.

Findings

Proved the Parisi formula for mixed p-spin models with odd p

Unified previous results for even p with new odd p cases

Confirmed the validity of the Parisi formula in broader models

Abstract

The Parisi formula for the free energy in the Sherrington-Kirkpatrick and mixed $p$-spin models for even $p\geq2$ was proved in the seminal work of Michel Talagrand [Ann. of Math. (2) 163 (2006) 221-263]. In this paper we prove the Parisi formula for general mixed $p$-spin models which also include $p$-spin interactions for odd $p$. Most of the ideas used in the paper are well known and can now be combined following a recent proof of the Parisi ultrametricity conjecture in [Ann. of Math. (2) 177 (2013) 383-393].