Mathematical Sciences: Research in Probability With Applications to Statistical Mechanics & Queueing Networks

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The principal investigator will conduct research in the area of large deviations. In particular, he will conduct projects on three topics: (a) large deviations for Markov chains taking values in an abstract space; (b) large deviations, limit theorems, and phase transitions in statistical mechanics; (c) large deviations for Markov processes with "discontinuous statistics" and applications to queueing networks. Large deviation theory is a branch of probability theory involving limits of random variables. It is well known that in certain idealized cases, where the random variables are independent and identically distributed, that the average of the sequence converges to the average of the individual random variable (in terms of statistics, that the sample average converges to the population average). Large deviation theory shows that the probability that the sample average deviates from the population average by a small amount is exponentially small as the sample size increases. The principal investigator is working in the much more difficult case of dependent random variables, whose dependence structure follows a Markov process. There are many applications of this theory to statistical mechanics and queueing networks.