## The Annals of Probability

- Ann. Probab.
- Volume 46, Number 2 (2018), 865-896.

### Free energy in the mixed $p$-spin models with vector spins

#### Abstract

Using the synchronization mechanism developed in the previous work on the Potts spin glass model, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand’s upper bound for the free energy of multiple mixed $p$-spin systems coupled by constraining their overlaps.

#### Article information

**Source**

Ann. Probab., Volume 46, Number 2 (2018), 865-896.

**Dates**

Received: April 2016

Revised: April 2017

First available in Project Euclid: 9 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1520586271

**Digital Object Identifier**

doi:10.1214/17-AOP1194

**Mathematical Reviews number (MathSciNet)**

MR3773376

**Zentralblatt MATH identifier**

06864075

**Subjects**

Primary: 60F10: Large deviations 60G15: Gaussian processes 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 $p$-spin interactions vector spins

#### Citation

Panchenko, Dmitry. Free energy in the mixed $p$-spin models with vector spins. Ann. Probab. 46 (2018), no. 2, 865--896. doi:10.1214/17-AOP1194. https://projecteuclid.org/euclid.aop/1520586271