Wolfgang König

Orthogonal polynomial ensembles in probability theory

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an {it orthogonal polynomial ensemble}. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. n nMuch attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area. n nIn the present text, we introduce various models, explain the questions and problems, and point out the relations between the models. Furthermore, we concisely outline some elements of the proofs of some of the most important results. This text is aimed at non-experts with strong background in probability who want to achieve a quick survey over the field.

The Universality Classes in the Parabolic Anderson Model

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on

Large deviations for trapped interacting Brownian particles and paths

mathbb{Z}^{d}

Non-Colliding Random Walks, Tandem Queues, and DiscreteOrthogonal Polynomial Ensembles

. We consider general i.i.d. potentials and show that exactly four qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at ∞ that are thicker than the double-exponential tails, (2) double-exponential tails at ∞ studied by Gärtner and Molchanov, (3) a new class called almost bounded potentials, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities. Our analysis of class (3) relies on two large deviation results for the local times of continuous-time simple random walk. One of these results is proved by Brydges and the first two authors in [BHK04], and is also used here to correct a proof in [BK01].

Annealed deviations of random walk in random scenery

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhans lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of Z d tending to Z d as time diverges. We finally discuss the relation of our density formula to the Ray-Knight theorem for continuous-time simple random walk on Z, which is analogous to the well-known Ray-Knight description of Brownian local times. In this extended version, we prove that the Ray-Knight theorem follows from our density formula.

Deviations of a random walk in a random scenery with stretched exponential tails

are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the pathrepellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose– Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross–Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross–Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.

Moments and Distribution of the Local Times of a Transient Random Walk on ℤd

AbstractA polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e−β for every self-intersection and eγ/(2d) for every self-contact to the probability of an n-step simple random walk on ℤd, where β,xa0γ>0 are parameters. It is known that for d=1 and γ>β the chain collapses down to finitely many sites, while for d=1 and γ<β it spreads out ballistically. Here we study for d=1 the critical case γ=β corresponding to the collapse transition and show that the end-to-end distance runs on the scale αn=n