The Aizenman-Sims-Starr scheme and Parisi formula for mixed $p$-spin spherical models

Abstract

The Parisi formula for the free energy in the spherical models with mixed even p-spin interactions was proven in Michel Talagrand [16]. In this paper we study the general mixed p-spin spherical models including p-spin interactions for odd p. We establish the Aizenman-Sims-Starr scheme and from this together with many wellknown results and Dmitry Panchenko's recent proof on the Parisi ultrametricity conjecture [11], we prove the Parisi formula.

Introduction to the SK model

“…We will discuss these proofs briefly in Section 12. By now, the analogues of the Parisi formula were proved for various modifications of the SK model: spherical SK model in Talagrand [91] (the formula was discovered by Crisanti and Sommers in [29], and another proof was given in Chen [21]); Ghatak-Sherrington model in Panchenko [65]; and multi-species SK model in Panchenko [77] (see Section 14 for discussion of this and related models).…”

“…Notice that, in the limit the array (R , ) , ≥1 would still be non-negative definitive and exchangeable in the sense of (21). Such arrays are called Gram-de Finetti arrays and the Dovbysh-Sudakov representation [37] guarantees that they can be generated as follows.…”

“…It is obvious that R κ is replica symmetric in distribution (or exchangeable) under EG ⊗∞ in the sense of the definition (21), and the Dovbysh-Sudakov representation in Theorem 1 yields that (we use here that the diagonal elements of R κ are equal to κ(q * ))…”

Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

“…(Mixed p-spin models) Mixed p-spin models on the sphere can be defined analogously to those on the hypercube (see Section 2.1). In this setting, the AGGIs still hold in the same regime as in the hypercube models (see, for example, [12,13]) and thus our methods still apply; however, much more is known about . For example, a direct computation shows that the Taylor expansion for…”

Free energy of multiple systems of spherical spin glasses with constrained overlaps

“…In [30], Talagrand proved a formula for the free energy of the spherical mixed even-p-spin model originally considered by Crisanti and Sommers in [9]. It was later extended to general mixed p-spin models by Chen in [4]. This formula is the analogue of the classical Parisi formula for the Sherrington-Kirkpatrick model [23,24] proved in [31].…”