Spectral properties of a tight binding Hamiltonian with period doubling potential

We study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence. We prove that its spectrum is purely singular continuous and supported on a Cantor set of zero Lebesgue measure, for all nonzero values of the potential strength. Moreover, we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength.     

Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if 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 provlem left open in previous work. 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.     

Rigorous results on the thermodynamics of the dilute Hopfield model

We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) 0, asN, 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 additionm(N) < ln N/ln 2, we prove that there exists, forT< 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 a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.     

Rigorous bounds on the storage capacity of the dilute Hopfield model

We study a neural network model consisting ofN neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1-p. For the Hopfield Hamiltonian on such a network, we prove that ifp c[(lnN)/N]^1/2, the model can store at leastm= _cpN patterns, where _c 0.027 ifc 3 and decreases proportional to 1/(–lnc) forc small. This generalizes the results of Newman for the standard Hopfield model.     

Metastability and Low Lying Spectra¶in Reversible Markov Chains

We study a large class of reversible Markov chains with discrete state space and transition matrix P _N . We define the notion of a set of metastable points as a subset of the state space Γ _N such that (i) this set is reached from any point x∈Γ _N without return to x with probability at least b _N , while (ii) for any two points x, y in the metastable set, the probability T ^{− 1} _{x , y} to reach y from x without return to x is smaller than a _{N − 1}< b _N . Under some additional non-degeneracy assumption, we show that in such a situation: (i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. (ii) To each metastable point corresponds one simple eigenvalue of 1 −P _N which is essentially equal to the inverse mean exit time from this state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution. Received: 1 August 2000 / Accepted: 19 November 2001     

Local Energy Statistics in Spin Glasses

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. We review some rigorous results confirming the validity of this conjecture. In the context of the SK models, we analyse the limits of the validity of the conjecture for energy levels growing with the volume of the system. In the case of the Generalised Random energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies E _N < β _c N, where β _c is the critical temperature. We also explain the more complex behaviour that sets in at higher energies. Research supported in part by the DFG in the Dutch-German Bilateral Research Group “Mathematics of Random Spatial Models from Physics and Biology” and by the European Science Foundation in the Programme RDSES.     

Poisson convergence in the restricted k‐partitioning problem

The randomized k‐number partitioning problem is the task to distribute N i.i.d. random variables into k groups in such a way that the sums of the variables in each group are as similar as possible. The restricted k‐partitioning problem refers to the case where the number of elements in each group is fixed to N/k. In the case k = 2 it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case k > 2 in the restricted problem and show that the vector of differences between the k sums converges to a k ‐ 1‐dimensional Poisson point process. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

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

Summary We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distribution of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695     

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

The low-temperature phase of Kac-Ising models

We analyze the low-temperature phase of ferromagnetic Kax-Ising models in dimensionsd2. We show that if the range of interactions is ^–1, then two disjoint translation-invariant Gibbs states exist if the inverse temperature satisfies –1^N, where =d(1–)/(2d+2)(d+1), for any >0. The proof involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous-spin system which is suitable for the use of a variant of the Peierls argument.