# Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model

@article{Guerra2002CentralLT, title={Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model}, author={Francesco Guerra and Fabio L. Toninelli University of Rome 'La Sapienza' and Infn and Rom{\'e} and Scuola Normale Superiore Pisa and Pisa}, journal={Journal of Mathematical Physics}, year={2002}, volume={43}, pages={6224-6237} }

In a region above the Almeida–Thouless line, where we are able to control the thermodynamic limit of the Sherrington–Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale 1/N, for large N. The method we employ is based on the idea we recently developed of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the…

## 29 Citations

### 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…

### The Ising--Sherrington-Kirpatrick Model in a Magnetic Field at High Temperature

- Physics
- 2005

We study a spin system on a large box with both Ising interaction and Sherrington-Kirpatrick couplings, in the presence of an external field. Our results are: (i) existence of the pressure in the…

### A Self-Averaging “Order Parameter” for the Sherrington-Kirkpatrick Spin Glass Model

- Mathematics
- 2004

Following an idea of van Enter and Griffiths, we define a self-averaging parameter for the Sherrington-Kirkpatrick (SK) spin glass which is a self-averaging version of the order parameter introduced…

### A Central Limit Theorem for Weighted Averages of Spins in the High Temperature Region of the Sherrington-Kirkpatrick Model

- Computer Science
- 2004

In this paper we prove that in the high temperature region of the Sherrington-Kirkpatrick model for a typical realization of the disorder the weighted average of spins $\sum_{i \leq N} t_i \sigma_i$…

### Fluctuations for the Bipartite Sherrington–Kirkpatrick Model

- MathematicsJournal of Statistical Physics
- 2021

We consider the bipartite Sherrington–Kirkpatrick model in which the variance of disorders depends only on the species the particles belong to. By using the stochastic calculus method, we obtain the…

### The replica symmetric region in the Sherrington-Kirkpatrick mean field spin glass model. The Almeida-Thouless line

- Physics
- 2006

In previous work, we have developed a simple method to study the behavior of the Sherrington-Kirkpatrick mean field spin glass model for high temperatures, or equivalently for high external fields.…

### 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…

### Higher Order Expansions for the Overlap of the SK Model

- Mathematics
- 2004

In this note, the Sherrington-Kirkpatrick model of interacting spins is under consideration. In the high temperature region, we give an asymptotic expansion for the expected value of some genereral…

### Higher order expansions for the overlap of the SK model

- Mathematics
- 2002

In this note, the Sherrington Kirkpatrick model of interacting spins is under consideration. In the high temperature region, we give an asymptotic expansion for the expected value of some genereral…

### A Central Limit Theorem for a Localized Version of the SK Model

- Mathematics, Physics
- 2006

In this note, we consider a SK (Sherrington–Kirkpatrick)-type model on ℤd for d≥1, weighted by a function allowing to any single spin to interact with a small proportion of the other ones. In the…

## References

SHOWING 1-10 OF 24 REFERENCES

### Replica symmetry breaking and exponential inequalities for the Sherrington-Kirkpatrick model

- Mathematics
- 2000

We provide an extremely accurate picture of the Sherrington-Kirkpatrick model in three cases: for high temperature, for large external field and for any temperature greater than or equal to 1 and…

### 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…

### Some rigorous results on the Sherrington-Kirkpatrick spin glass model

- Physics
- 1987

We prove that in the high temperature regime (T/J>1) the deviation of the total free energy of the Sherrington-Kirkpatrick (S-K) spin glass model from the easily computed log Av(Z
N
({βJ}))…

### The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case

- Mathematics
- 1995

We study the fluctuations of free energy, energy and entropy in the high temperature regime for the Sherrington-Kirkpatrick model of spin glasses. We introduce here a new dynamical method with the…

### Some comments on the Sherrington-Kirkpatrick model of spin glasses

- Physics
- 1987

In this paper the high-temperature phase of general mean-field spin glass models, including the Sherrington-Kirkpatrick (SK) model, is analyzed. The free energy in zero magnetic field is calculated…

### 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…

### The Cavity Method In The Mean Field Spin Glass Model. Functional Representations Of Thermodynamic Va

- Mathematics
- 1994

We consider the Sherrington-Kirkpatrick mean eld model for spin glasses and show how the cavity method can be exploited for constructing functional representations of the ther-modynamic variables, in…

### 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…

### On the Stability of the Quenched State in Mean-Field Spin-Glass Models

- Mathematics
- 1998

While the Gibbs states of spin-glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying “quenched state.”…

### ABOUT THE OVERLAP DISTRIBUTION IN MEAN FIELD SPIN GLASS MODELS

- Physics
- 1996

We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps…