## The Annals of Probability

- Ann. Probab.
- Volume 35, Number 6 (2007), 2321-2355.

### On the overlap in the multiple spherical SK models

Dmitry Panchenko and Michel Talagrand

#### Abstract

In order to study certain questions concerning the distribution of the overlap in Sherrington–Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with constrained overlaps. One can write analogues of Guerra’s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds. We were able to make some progress for spherical pure *p*-spin SK models where many computations can be made explicitly. For pure 2-spin model we prove ultrametricity and chaos in an external field. For the pure *p*-spin model for even *p*>4 without an external field we describe two possible values of the overlap of two systems at different temperatures. We also prove a somewhat unexpected result which shows that in the 2-spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and, for example, ultrametricity holds only in a weak sense.

#### Article information

**Source**

Ann. Probab., Volume 35, Number 6 (2007), 2321-2355.

**Dates**

First available in Project Euclid: 8 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1191860423

**Digital Object Identifier**

doi:10.1214/009117907000000015

**Mathematical Reviews number (MathSciNet)**

MR2353390

**Zentralblatt MATH identifier**

1128.60086

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

**Keywords**

Spin glasses free energy chaos ultrametricity

#### Citation

Panchenko, Dmitry; Talagrand, Michel. On the overlap in the multiple spherical SK models. Ann. Probab. 35 (2007), no. 6, 2321--2355. doi:10.1214/009117907000000015. https://projecteuclid.org/euclid.aop/1191860423