Spin glass models from the point of view of spin distributions

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 $\sigma$.

Published in at http://dx.doi.org/10.1214/11-AOP696 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• On exchangeable random variables and the statistics of large graphs and hypergraphs
• The Ghirlanda-Guerra identities without averaging
• Random Overlap Structures: Properties and Applications to Spin Glasses

Cited by in corpus (33)

• Information-theoretic thresholds from the cavity method
• The Parisi formula for mixed $p$-spin models
• Approximate Ultrametricity for Random Measures and Applications to Spin Glasses
• On the TAP free energy in the mixed $p$-spin models
• Harnessing the Bethe free energy
• A connection between MAX $\kappa$-CUT and the inhomogeneous Potts spin glass in the large degree limit
• Structure of finite-RSB asymptotic Gibbs measures in the diluted spin glass models
• The generalized TAP free energy II
• The marginally stable Bethe lattice spin glass revisited
• Limits of discrete distributions and Gibbs measures on random graphs
• A unified stability property in spin glasses
• Structure of 1-RSB asymptotic Gibbs measures in the diluted p-spin models
• Spin systems on Bethe lattices
• The random 2-SAT partition function
• On spin distributions for generic p-spin models
• Short-range spin glasses and Random Overlap Structures
• Exchangeable random measures
• The full replica symmetry breaking in the Ising spin glass on random regular graph
• Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures
• Replica Symmetry Breaking without Replicas
• On the free energy of vector spin glasses with non-convex interactions
• A hierarchical version of the de Finetti and Aldous-Hoover representations
• The rank of random matrices over finite fields
• On the $K$-sat model with large number of clauses
• Limit theorems for invariant distributions
• Free energy of a diluted spin glass model with quadratic Hamiltonian
• Tailoring the overlap distribution in driven mean-field spin models
• On the replica symmetric solution of the K-sat model
• Replica bound for Ising spin glass models in one dimension
• Universality of Ghirlanda-Guerra identities and spin distributions in mixed $p$-spin models
• The full replica symmetry breaking solution in mean-field spin glass models
• The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models
• On the replica symmetry phase of the independent set problem