# Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

@article{Guerra2003BrokenRS, title={Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model}, author={Francesco Guerra}, journal={Communications in Mathematical Physics}, year={2003}, volume={233}, pages={1-12} }

Abstract: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from…

## 306 Citations

### Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model

- Mathematics
- 2009

We give a short review about recent results in the study of the mean field Sherrington-Kirkpatrick model for a spin glass. Our methods are essentially based on interpolation and comparison arguments…

### The full replica symmetry breaking solution in mean-field spin glass models

- Mathematics
- 2019

This thesis focus on the extension of the Parisi full replica symmetry breaking solution to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the…

### Multi-Species Mean Field Spin Glasses. Rigorous Results

- Mathematics
- 2013

We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington–Kirkpatrick one for the single species case. The…

### On a dynamical-like replica-symmetry-breaking scheme for the spin glass

- Physics
- 2004

Abstract.Considering the unphysical result obtained in the calculation of the free-energy cost for twisting the boundary conditions in a spin glass, we trace it to the negative multiplicities…

### On the mean-field spin glass transition

- Physics
- 2008

In this paper we analyze two main prototypes of disordered mean-field systems, namely the Sherrington-Kirkpatrick (SK) and the Viana-Bray (VB) models, to show that, in the framework of the cavity…

### Exact Solution of the Gauge Symmetric p-Spin Glass Model on a Complete Graph

- Physics
- 2009

We consider a gauge symmetric version of the p-spin glass model on a complete graph. The gauge symmetry guarantees the absence of replica symmetry breaking and allows to fully use the interpolation…

### The High Temperature Region of the Viana–Bray Diluted Spin Glass Model

- Mathematics
- 2003

In this paper, we study the high temperature or low connectivity phase of the Viana–Bray model in the absence of magnetic field. This is a diluted version of the well known Sherrington–Kirkpatrick…

### Interpolation and Comparison Methods in the Mean Field Spin Glass Model

- Mathematics
- 2014

We give a short overview of the recent rigorous mathematical methods developed for the study of complex disordered systems, in particular spin glasses in the mean field Sherrington-Kirkpatrick…

### Twist free energy in a spin glass

- Physics
- 2002

Abstract:The field theory of a short range spin glass with Gaussian random interactions, is considered near the upper critical dimension six. In the glassy phase, replica symmetry breaking is…

### The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings

- MathematicsArXiv
- 2021

This analysis is an adaption of a “conditional quenched equals annealed” argument used by Bolthausen to analyze the high-temperature regime of the Sherrington-Kirkpatrick model.

## References

SHOWING 1-10 OF 19 REFERENCES

### The Thermodynamic Limit in Mean Field Spin Glass Models

- Mathematics
- 2002

Abstract: We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as…

### Fluctuations and thermodynamic variables in mean field spin glass models

- Physics
- 1992

We present two rigorous results on the Sherrington-Kirkpatrick mean field model for spin glasses, proven by elementary methods, based on properties of fluctuations, with respect to the external…

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

- Physics
- 2002

We develop a very simple method to study the high temperature, or equivalently high external field, behavior of the Sherrington–Kirkpatrick mean field spin glass model. The basic idea is to couple…

### Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)

- Mathematics
- 2000

Abstract:The mean field spin glass model is analyzed by a combination of exact methods and a simple Ansatz. The method exploited is general, and can be applied to others disordered mean field models…

### Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

- Mathematics
- 1991

AbstractWe prove that ifĤNis the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity
$$\bar q_N = N^{ - 1} \sum \left\langle {S_l } \right\rangle _H^2 $$
converges in the variance to a…

### Infinite Volume Limit and Spontaneous Replica Symmetry Breaking in Mean Field Spin Glass Models

- Mathematics
- 2003

Mean field spin glasses [1] were introduced around 30 years ago as an approximation of realistic models for disordered magnetic alloys. In what follows, we make particular reference to the well known…

### About the Almeida-Thouless transition line in the Sherrington-Kirkpatrick mean-field spin glass model

- Physics
- 2002

We consider the Sherrington-Kirkpatrick model and we prove that the quenched free energy per spin is strictly larger than the corresponding replica symmetric approximation, for all values of the…

### A sequence of approximated solutions to the S-K model for spin glasses

- Physics
- 1980

In the framework of the new version of the replica theory, a sequence of approximated solutions is computed for the Sherrington-Kirkpatrick model (see Phys. Rev. Lett., vol.35, p.1972, 1975) of spin…