Self-averaging in a class of generalized Hopfield models

We prove the almost sure convergence to zero of the fluctuations of the free energy, resp. local free energies, in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: 1. multi-spin interactions are permitted and 2. the random variables #xi#_i"#mu# describing the 'patterns' can have arbitrary distributions with mean zero and finite 4+#epsilon#-th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven. (orig.)Available from TIB Hannover: RR 5549(105) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

A rigorous renormalization group method for interfaces in random media

We prove the existence Gibbs states describing rigid interfaces in a disordered solid-on-solid (SOS) for low temperatures and for weak disorder in dimension D#>=#4. This extends earlier results for hierarchical models to the more realistic models and proves a long-standing conjecture. The proof is based on the renormalization group method of Bricmont and Kupiainen originally developed for the analysis of low-temperature phases of the random field Ising model. In a broader context, we generalize this method to a class of systems with non-compact single-site state space. (orig.)Available from TIB Hannover: RR 5549(45)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Mathematical physics of disordered systems Abstracts

SIGLEAvailable from TIB Hannover: RR 3285(3)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Rigorous results on the Hopfield model of neural networks

We review some recent rigorous results in the theory of neural networks, and in particular on the thermodynamic properties of the Hopfield model. In this context, the model is treated as a Curie-Weiss model with random interactions and large deviation techniques are applied. The tractability of the random interactions depends strongly on how the number, M, of stored patterns scales with the size, N, of the system. We present an exact analysis of the thermodynamic limit under the sole condition that M/N #arrow down# 0, as N #arrow up# #infinity#, i.e. we prove the almost sure convergence of the free energy to a non-random limit and the a.s. convergence of the measures induced on the overlap parameters. We also present results on the structure of local minima of the Hopfield Hamiltonian, originally derived by Newman. All these results are extended to the Hopfield model defined on dilute random graphs. (orig.)Available from TIB Hannover: RR 5549(60)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Rigorous results on the thermodynamics of the dilute Hopfield model

We study the Hopfield model of an autoassociative memory on a random graph on N vertices where the probability of two vertices being joined by a link is p(N). Assuming that p(N) goes to zero more slowly than O(1/N), we prove the following results: 1. If the number of stored patterns, m(N), is small enough such that m(N)/(Np(N)) #arrow down# 0, as N #arrow up# #infinity#, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. 2. If in addition m(N) #<=#N/ln2, we prove that there exists, for T #<=# 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is an uniform bound on the difference between the Hamiltonian on a random graph and its mean value. (orig.)Available from TIB Hannover: RR 5549(19)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

The thermodynamics of the Curie-Weiss model with random couplings

We study the Curie-Weiss version of an Ising spin system with random, positively biased, couplings. In particular the case where the couplings #epsilon#_i_j take the values one with probability p and zero with probability 1 - p which describes the Ising model on a random graph is considered. We prove that if p is allowed to decrease with the system size N in such a way that N p(N)#arrow up# #infinity# as N #arrow up# #infinity#, then the free energy converges (after trivial rescaling) to that of the standard Curie Weiss model, almost surely. Equally, the induced measure on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed. (orig.)Available from TIB Hannover: RR 5549(6)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Large deviation principles for the Hopfield model and the Kac-Hopfield model

We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distributions of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns. (orig.)Available from TIB Hannover: RR 5549(95)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model with M(N)=#alpha#N patterns, where N is the number of neurons. We show that if #alpha# is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work [BGP1]. The key new ingredient is a self averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given. (orig.)Available from TIB Hannover: RR 5549(97)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Spectral properties of one-dimensional Schroedinger operators with potentials generated by substitutions

We investigate one-dimensional discrete Schroedinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable hypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schroedinger operator is singular und supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples. (orig.)Available from TIB Hannover: RR 5549(4)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Remarks on the spectral properties of tight binding and Kronig-Penney models with substitution sequences

We comment on some recent investigations on the electronic properties of models associated to the Thue-Morse chain and point out that their conclusions are in contradiction with rigorously proven theorems and indicate some of the sources of these misinterpretations. We briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems. (orig.)Available from TIB Hannover: RR 5549(82)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman