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**Rigorous results for the Hopfield model with many patterns.**
*(English)*
Zbl 0897.60041

Two models of disordered systems have been the focus of attention for theoretical and mathematical physics over the last two decades: the Sherrington-Kirkpatrick model and the Hopfield model. In the Hopfield model in particular, in various regimes of the parameters of the model (in particular the temperature \(\beta^{-1}\) and the number of the stored patterns (or more precisely the ratio \(\alpha\), of the number of stored patterns \(M(N)\) over the volume \(N\) of the system)) simplifications occur that have allowed to make progress in the rigorous understanding of this system. In this article a major breakthrough is achieved towards a rigorous justification of the results of the so called “replica approach”, a heuristic technique which is used heavily in the physics literature to study disordered systems that has, however, never been put on a mathematically rigorous basis. Besides that, a number of previous results are recovered or improved. The paper is organized along the different regimes of the model. Section 2 deals with the high-temperature regime, more precisely the regime \(\beta<1/(1+\sqrt\alpha)\). Here a simple and elegant proof of the equality of the quenched and annealed free energy is given, along with sharp bounds on the fluctuations of the free energy.

Section 3 deals with the low-temperature regime for small \(\alpha\). The main result shown here is that the Gibbs measures are concentrated on small balls around the stored patterns in the space of the “overlap-parameters”. These results are the basic input for what is to follow in the subsequent sections. They are very similar to estimates obtained by the reviewer and V. Gayrard [Probab. Theory Relat. Fields 107, No. 1, 61-98 (1997; 866.60085)] and sharpen the earliest results of that nature obtained by the reviewer, V. Gayrard and P. Picco [ibid. 100, No. 3, 329-363 (1994; Zbl 0810.60094) and J. Stat. Phys. 79, 395-414 (1995)].

Sections 4-8 form the core of the paper and contain the main new results. Roughly speaking, they establish a non-trivial regime of parameters (in particular \(\alpha>0\), and \(\beta>1\)) in which it is proven that the formal solution of the model obtained by the replica method by Amit, Gutfreund and Sompolinsky [Ann. Phys. 173, 30-67 (1987)] is correct. This includes in particular an exact expression for the free energy and the expected value of the overlap parameters. The method of proof here is induction over the volume assisted by the a priori estimates from Section 3. This idea as such is not new. Under the name of “cavity method” it was used in the spin glass theory as an alternative approach to the replica method [see Mézard et al., “Spin glass theory and beyond” (World Scientific, Singapore, 1987)] in a formal way (i.e. without control of error terms). A first attempt to put this method on a rigorous basis in the context of the Hopfield model was undertaken by L. Pastur, M. Shcherbina and B. Tirozzi [J. Stat. Phys. 74, No. 5/6, 1161-1183 (1994; Zbl 0835.60092)]. The present paper, however, is the first to use this method to give a fully rigorous analysis of this model.

The final section concerns the zero-temperature results, or what is commonly called as the “storage capacity” problem. Here the issue is for which values of \(\alpha\) the stored patterns are local minima of the Hamiltonian function. Numerical results and replica results indicate a critical value \(a_c\) of order \(0.14\). Here rigorous bounds of Newman [Neural Networks 1, 223-238 (1988)] and their recent improvements by D. Loukianova [C. R. Acad. Sci., Paris, Sér. I 318, 157-160 (1994; Zbl 0794.92002)] are numerically improved. There has already been considerable progress since this paper was written and a good review of this development can be found in the author’s contribution to the Proceedings of the ICM ’98 [to appear in Acta Math., 1998].

Section 3 deals with the low-temperature regime for small \(\alpha\). The main result shown here is that the Gibbs measures are concentrated on small balls around the stored patterns in the space of the “overlap-parameters”. These results are the basic input for what is to follow in the subsequent sections. They are very similar to estimates obtained by the reviewer and V. Gayrard [Probab. Theory Relat. Fields 107, No. 1, 61-98 (1997; 866.60085)] and sharpen the earliest results of that nature obtained by the reviewer, V. Gayrard and P. Picco [ibid. 100, No. 3, 329-363 (1994; Zbl 0810.60094) and J. Stat. Phys. 79, 395-414 (1995)].

Sections 4-8 form the core of the paper and contain the main new results. Roughly speaking, they establish a non-trivial regime of parameters (in particular \(\alpha>0\), and \(\beta>1\)) in which it is proven that the formal solution of the model obtained by the replica method by Amit, Gutfreund and Sompolinsky [Ann. Phys. 173, 30-67 (1987)] is correct. This includes in particular an exact expression for the free energy and the expected value of the overlap parameters. The method of proof here is induction over the volume assisted by the a priori estimates from Section 3. This idea as such is not new. Under the name of “cavity method” it was used in the spin glass theory as an alternative approach to the replica method [see Mézard et al., “Spin glass theory and beyond” (World Scientific, Singapore, 1987)] in a formal way (i.e. without control of error terms). A first attempt to put this method on a rigorous basis in the context of the Hopfield model was undertaken by L. Pastur, M. Shcherbina and B. Tirozzi [J. Stat. Phys. 74, No. 5/6, 1161-1183 (1994; Zbl 0835.60092)]. The present paper, however, is the first to use this method to give a fully rigorous analysis of this model.

The final section concerns the zero-temperature results, or what is commonly called as the “storage capacity” problem. Here the issue is for which values of \(\alpha\) the stored patterns are local minima of the Hamiltonian function. Numerical results and replica results indicate a critical value \(a_c\) of order \(0.14\). Here rigorous bounds of Newman [Neural Networks 1, 223-238 (1988)] and their recent improvements by D. Loukianova [C. R. Acad. Sci., Paris, Sér. I 318, 157-160 (1994; Zbl 0794.92002)] are numerically improved. There has already been considerable progress since this paper was written and a good review of this development can be found in the author’s contribution to the Proceedings of the ICM ’98 [to appear in Acta Math., 1998].

Reviewer: A.Bovier (Berlin)

### MSC:

60G15 | Gaussian processes |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

60E15 | Inequalities; stochastic orderings |