Agmon-type estimates for a class of jump processes

In the limit epsilon to 0 we analyze the generators H_epsilon of families of reversible jump processes in R^d associated with a class of symmetric non-local Dirichlet-forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being C^2 or just Lipschitz. Our estimates are analog to the semi-classical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice epsilon Z^d. Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques

Metastable states, quasi-stationary distributions and soft measures

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypothesis for (families of) Markov chains on finite configuration space in some asymptotic regime, including the case of configuration space size going to infinity. By comparing restricted ensemble and quasi-stationary measures, we study point-wise convergence velocity of Yaglom limits and prove asymptotic exponential exit law. We introduce soft measures as interpolation between restricted ensemble and quasi-stationary measure to prove an asymptotic exponential transition law on a generally different time scale. By using potential theoretic tools, we prove a new general Poincar\'e inequality and give sharp estimates via two-sided variational principles on relaxation time as well as mean exit time and transition time. We also establish local thermalization on a shorter time scale and give mixing time asymptotics up to a constant factor through a two-sided variational principal. All our asymptotics are given with explicit quantitative bounds on the corrective terms.Comment: 41 page

An almost sure large deviation principle for the Hopfield model

We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random patterns, $M$, as a function of the system size $N$ satisfies $\limsup M(N)/N=0$. In this case the rate function (or free energy as a function of the overlap parameters) is independent of the disorder for almost all realization of the patterns and given by an explicit variational formula.Comment: 31pp; Plain-TeX, hardcopy available on request from [email protected]

Spectral characterization of aging: the rem-like trap model

We review the aging phenomenon in the context of the simplest trap model, Bouchaud's REM-like trap model, from a spectral theoretic point of view. We show that the generator of the dynamics of this model can be diagonalized exactly. Using this result, we derive closed expressions for correlation functions in terms of complex contour integrals that permit an easy investigation into their large time asymptotics in the thermodynamic limit. We also give a ``grand canonical'' representation of the model in terms of the Markov process on a Poisson point process. In this context we analyze the dynamics on various time scales.Comment: Published at http://dx.doi.org/10.1214/105051605000000359 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Retrieval Phase of the Hopfield Model: A Rigorous Analysis of the Overlap Distribution

Standard large deviation estimates or the use of the Hubbard-Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function \Phi_{N,\b} on $\R^M$. In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m^*(\b) the modulus of the spontaneous magnetization in the Curie-Weiss model and by \a the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0<\g_a<\g_c such that 1) If \sqrt\a\leq \g_a (m^*(\b))^2, then the absolute minima of $\Phi$ are located within small balls around the points $\pm m^*e^\mu$, where $e^\mu$ denotes the $\mu$-th unit vector while 2) if \sqrt\a\leq \g_c (m^*(\b))^2 at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures. KEYWORDS: Hopfield model, neural networks, storage capacity, Gibbs measures, self-averaging, random matricesComment: 43 pages, uuencoded, Z-compressed Postscrip

Poisson convergence in the restricted kk-partioning 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.Comment: 31pp, AMSTe