Max independence structures and Tracy-Widom
A talk in the Bielefeld-Melbourne Seminar series by
Yacine Barhoumi
| 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 Schur 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 GUEN
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. Last, we will show that the Tracy-Widom distribution
is also a maximum of an infinite number of independent random variables. Within the CRC this talk is associated to the project(s): B5, C6 |