Research on classical and non-commutative probability

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project involves research on conditional moments of random fields, random matrices, noncommutative probability, and large deviations.The PI will explore connections between classical and free probability from several inter-related angles. To reconcile the classical and free central limit theorems, he will study a noncommutative central limit theorem that was suggested by examples in his previous work. He will study similarities between the exponential families of statistics and the Cauchy-Stieltjes kernel families as a way to relate the normal, binomial, gamma and Poisson laws to their counterparts in free probability. He will investigate matrix ensembles that share similarities with such laws, and may lead to new matrix models for the corresponding free probability laws. The PI will use the orthogonality measures of the Askey-Wilson polynomials as a replacement for the above mentioned classical laws to construct Markov processes with linear regressions and quadratic conditional variances on an interval.He will also analyze large deviations of Markov chains that describe the behavior of geometric quantities like the number of vertexes of prescribed degree as they vary during the evolution of a random tree.This research originated from the study of random processes that have linear regressions and quadratic conditional variances. These processes model phenomena that evolve at random when the initial and final endpoints are given by following a straight line on average. The randomness occurs as deviations from that line with variance that is a quadratic function of the endpoints. Such processes, not surprisingly, turn out to be Markov; but surprisingly they exhibit intimate connections to noncommutative probability, and in particular to free probability that usually arises as approximation to spectra of large random matrices. Thus the PI will also investigate random matrices, quadratic regression problems, and the centrial limit theorem in a noncommutative setting. A separate topic to be investigated are rare phenomena arising in random tree models that evolve in time. Random trees serve as models in biology, psychology, and computer science.Rare events of interest consist of unusually large deviations from the equilibrium, and they model a rare but influential behavior of the system.