Carla Mariana da Silva Pinheiro

Random matrix theory and some results for the Painleve-I kernel

Applications of Random Matrix Theory can be found everywhere. From nuclear resonance to machine learning and even natural phenomena are some examples of systems whose behavior is predicted by eigenvalues in random matrix ensembles. In unitary ensembles the eigenvalues constitute a DPP and the relevant statistics can be obtained through the study of the correspondent kernel. In this context, it is well known that when the density of the equilibrium measure has a “soft edge” (behaves as x^(1/2)), the associated kernel converges to the Airy kernel. And, as stated in a 2010 paper by Claeys-Its-Krasovsky, as the edge becomes “softer” (behavior of order x^(5/2), x^(9/2) and so on), the limit kernel is given by means of solutions to the Painleve-I hierarchy.

The present seminar is divided into two parts. Firstly, we present (in a very non-technical way) some applications and standard techniques that motivates the studies of kernels arising in RMT as well as deformations of such kernels. Next, we present briefly some original results for a deformed higher-order Painleve-I kernel.