Equilibrium statistical mechanics of bipartite spin systems

@article{Barra2011EquilibriumSM,
title={Equilibrium statistical mechanics of bipartite spin systems},
author={Adriano Barra and Giuseppe Genovese and Francesco Guerra},
journal={Journal of Physics A},
year={2011},
volume={44},
pages={245002}
}
• Published 2011
• Mathematics, Physics
• Journal of Physics A
The aim of this paper is to give an extensive treatment of bipartite mean field spin systems, pure and disordered. At first, bipartite ferromagnets are investigated, and an explicit expression for the free energy is achieved through a new minimax variational principle. Then, via the Hamilton?Jacobi technique, the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one. The same is… Expand
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References

SHOWING 1-10 OF 49 REFERENCES
The Thermodynamic Limit in Mean Field Spin Glass Models
• Mathematics, Physics
• 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, asExpand
Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
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-KirkpatrickExpand
Replica symmetry breaking in mean-field spin glasses through the Hamilton–Jacobi technique
• Physics, Mathematics
• 2010
During the last few years, through the combined effort of the insight coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main featuresExpand
Irreducible Free Energy Expansion and Overlaps Locking in Mean Field Spin Glasses
Following the works of Guerra, 1995; Aizenmar and Contucci, J. State. Phys. 92 (5–6): 765–783 (1998), we introduce a diagrammatic formulation for a cavity field expansion around the criticalExpand
On the Stability of the Quenched State in Mean-Field Spin-Glass Models
• Mathematics, Physics
• 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.”Expand
Course 5 – An Introduction to Mean Field Spin Glas Theory: Methods and Results
The chapter introduces the ferromagnetic model, and discusses the behavior and properties of the free energy in the thermodynamic limit, in this very elementary case, the comparison and interpolationExpand
The Cavity Method In The Mean Field Spin Glass Model. Functional Representations Of Thermodynamic Va
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, inExpand
On the structure of quasi-stationary competing particle systems
• Mathematics, Physics
• 2009
We study point processes on the real line whose configurations X are locally finite, have a maximum and evolve through increments which are functions of correlated Gaussian variables. TheExpand
Bipartite Mean Field Spin Systems. Existence and Solution
• Physics, Mathematics
• 2007
A mean field spin system consisting two interacting groups each with homogeneous interaction coefficients is introduced and studied. Existence of the thermodynamic limit is shown by an asymptoticExpand
The Replica Symmetric Approximation of the Analogical Neural Network
• Physics, Mathematics
• 2010
In this paper we continue our investigation of the analogical neural network, by introducing and studying its replica symmetric approximation in the absence of external fields. Bridging the neuralExpand