264 research outputs found
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 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. Much
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.Comment: Published at http://dx.doi.org/10.1214/154957805100000177 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Reflected random walks and unstable Martin boundary
We introduce a family of two-dimensional reflected random walks in the
positive quadrant and study their Martin boundary. While the minimal boundary
is systematically equal to a union of two points, the full Martin boundary
exhibits an instability phenomenon, in the following sense: if some parameter
associated to the model is rational (resp. non-rational), then the Martin
boundary is discrete, homeomorphic to (resp. continuous,
homeomorphic to ). Such instability phenomena are very rare in the
literature. Along the way of proving this result, we obtain several precise
estimates for the Green functions of reflected random walks with escape
probabilities along the boundary axes and an arbitrarily large number of
inhomogeneity domains. Our methods mix probabilistic techniques and an analytic
approach for random walks with large jumps in dimension two.Comment: 46 pages, 7 figure
Random walks in the quarter plane, discrete harmonic functions and conformal mappings
With an appendix by Sandro Franceschi. 32 pages, 9 figures.International audienceWe propose a new approach for finding discrete harmonic functions in the quarter plane with Dirichlet conditions. It is based on solving functional equations that are satisfied by the generating functions of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time for random walks from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the discrete harmonic function
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