## The Annals of Probability

- Ann. Probab.
- Volume 46, Number 2 (2018), 829-864.

### Free energy in the Potts spin glass

#### Abstract

We study the Potts spin glass model, which generalizes the Sherrington–Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product, which includes the Potts spin glass as a special case. As another example of application, one can show that Talagrand’s bound for multiple copies of the mixed $p$-spin model with constrained overlaps is asymptotically sharp. We will consider these problems in the subsequent paper and illustrate the main new idea on the technically more transparent case of the Potts spin glass.

#### Article information

**Source**

Ann. Probab., Volume 46, Number 2 (2018), 829-864.

**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/1520586270

**Digital Object Identifier**

doi:10.1214/17-AOP1193

**Mathematical Reviews number (MathSciNet)**

MR3773375

**Zentralblatt MATH identifier**

06864074

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 60F10: Large deviations 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

**Keywords**

Spin glasses Sherrington–Kirkpatrick model Potts spin glass free energy

#### Citation

Panchenko, Dmitry. Free energy in the Potts spin glass. Ann. Probab. 46 (2018), no. 2, 829--864. doi:10.1214/17-AOP1193. https://projecteuclid.org/euclid.aop/1520586270