Alexandre Krajenbrink - Fredholm determinants, Painlevé equations and the Zakharov--Shabat system

As Fredholm determinants are more and more frequent in the context of stochastic integrability, I discuss in this talk the existence of a common framework in many integrable systems where they appear. This consists in a hierarchy of equations, akin to the Zakharov-Shabat system, connecting an integro-differential extension of the Painlevé II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation and multi-critical fermions at finite temperature.
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Integrability and Combinatorics at Finite Temperature 2021. Program at MATRIX Institute