Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model

Abstract

We develop a simple method to study the high temperature, or high external field, behavior of the Sherrington-Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we prove the validity of the Sherrington-Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida-Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi Ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be much simpler, and more tractable. By applying the cavity method, we show also how to determine free energy and overlap fluctuations, in the region where replica symmetry has been shown to hold.

17 pages; added references; corrected typos

References in corpus (2)

• About the overlap distribution in mean field spin glass models
• Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model

Cited by in corpus (29)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• A replica-coupling approach to disordered pinning models
• The high temperature region of the Viana-Bray diluted spin glass model
• Analysis of Regularized LS Reconstruction and Random Matrix Ensembles in Compressed Sensing
• A note on classical ground state energies
• Some Properties of the Phase Diagram for Mixed $p$-Spin Glasses
• Replica symmetry breaking in mean field spin glasses trough Hamilton-Jacobi technique
• Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model
• Sharp Bounds for Optimal Decoding of Low Density Parity Check Codes
• Deep Boltzmann machines: rigorous results at arbitrary depth
• Spin Glass Computations and Ruelle's Probability Cascades
• Interpolating the Sherrington-Kirkpatrick replica trick
• Replica symmetry breaking in multi-species Sherrington-Kirkpatrick model
• On the irrelevant disorder regime of pinning models
• Fluctuation results for Multi-species Sherrington-Kirkpatrick model in the replica symmetric regime
• Variational Bounds for the Generalized Random Energy Model
• A Remark on the Infinite-Volume Gibbs Measures of Spin Glasses
• Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment
• PDE/statistical mechanics duality: relation between Guerra's interpolated $p$-spin ferromagnets and the Burgers hierarchy
• Concentration of multi-overlaps for random ferromagnetic spin models
• Notes on the polynomial identities in random overlap structures
• A Solvable Mean Field Model of a Gaussian Spin Glass
• The replica symmetric region in the Sherrington-Kirkpatrick mean field spin glass model. The Almeida-Thouless line
• Mismatched Estimation of rank-one symmetric matrices under Gaussian noise
• The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature
• Mean Field Spin Glass Models under Weak External Field
• Applications of correlation inequalities to low density graphical codes
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• High temperature Sherrington-Kirkpatrick model for general spins