Applications of Random Matrix Theory can be found in almost every research area. 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 Determinantal Point Process and, consequently, the relevant statistics can be obtained through the study of the correspondent reproducing kernel. In this context, it is well known that when the equilibrium measure related to the probability measure on the space of matrices has a “soft edge” (behaves as x1/2), the associated kernel converges to the Airy kernel. And, as stated in [1], as the edge becomes “softer” (behavior of order x5/2, x9/2 and so on), the limit kernel is given by means of solutions to the Painlevé I hierarchy. The present seminar is divided into two parts. Firstly, we present 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 Painlevé I kernel.
