## The Annals of Probability

### Spin glass models from the point of view of spin distributions

Dmitry Panchenko

#### Abstract

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous–Hoover representation encoded by a function $\sigma : [0,1]^{4}\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington–Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington–Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda–Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.

#### Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1315-1361.

Dates
First available in Project Euclid: 29 April 2013

https://projecteuclid.org/euclid.aop/1367241501

Digital Object Identifier
doi:10.1214/11-AOP696

Mathematical Reviews number (MathSciNet)
MR3098679

Zentralblatt MATH identifier
1281.60081

#### Citation

Panchenko, Dmitry. Spin glass models from the point of view of spin distributions. Ann. Probab. 41 (2013), no. 3A, 1315--1361. doi:10.1214/11-AOP696. https://projecteuclid.org/euclid.aop/1367241501

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