Eigenvalues from random matrices offer a good model for several systems: from abstract objects such as log-gas and big-data to everyday-life elements such as coffee stains and buses timetables. In unitary ensembles, the eigenvalues constitute a determinantal point process and the relevant statistics are obtained through a kernel. In this context, when the density of the associated equilibrium measure vanishes as $x^{1/2}$ in the edge of the support, the kernel converges to the Airy one. And, it had been conjectured that for a vanishing order $x^{(4k+1)/2}$ one recovers the higher order Painlevé I kernel [1].
The present seminar brings some results in the context of a deformation of the higher order Painlevé I kernel. Firstly, some applications and motivation for the study of random matrix theory are presented in a very non-technical way. Then, the main definitions and standard techniques are stated. Finally, we discuss the results in the asymptotic characterization for integrable hierarchies obtained from the deformed kernel, inspired by [2]. The presentation is based in two joint works with Cafasso, one available on Arxiv:2504.20721 and one currently in preparation for submission.
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