The Annals of Probability

Spin glass models from the point of view of spin distributions

Dmitry Panchenko

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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 $\sigma$.

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

First available in Project Euclid: 29 April 2013

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Zentralblatt MATH identifier

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.)

Mean-field spin glass models perturbations stability


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.

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