Yacine Barhoumi-Andréani: Max-independence structures in random matrix theory and random partitions

Abstract:

We consider two classical and extensively studied models of random objects: eigenvalues of a GUE random matrix and random integer partitions distributed according to the Poisson-Plancherel measure. We express the largest element of these random sets as maxima of independent random variables. We then proceed to rescale the largest eigenvalue of the GUE_N written as a maximum of N independent random variables with the classical Poisson approximation for sums of indicators. We use for this the Okamoto-Noumi-Yamada theory of the sigma-form of the Painlevé equation applied to random matrix theory by Forrester-Witte (we will recall part of this theory). By doing so, we find a new expression for the cumulative distribution function of the GUE Tracy-Widom distribution which is shown to be equivalent to the classical one using manipulations à la Forrester-Witte. We will then highlight the mechanism that allows to get the largest part of a Poisson-Plancherel-distributed random partition as an i.i.d. randomisation of the sequence of negative integers. We will also give a general criteria for this type of structure to obtain the Tracy-Widom distribution at the limit using a local CLT for the law generating the i.i.d. sequence. If time permits, we will show that the GUE Tracy-Widom distribution is also a maximum of an infinite number of independent random variables whose law involves limiting versions of prolate hyperspheroidal wave functions, an optical “cousin” of the prolate spheroids related to the sine kernel operator.