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Thomas Chouteau. On a discrete Painlevé II hierarchy via orthogonal polynomials: Lax pair and application to multicritical random partitions.
In a recent article, Betea, Bouttier and Walsh presented a relation between higher order analogue of Tracy-Widom distribution and certain Toeplitz determinants describing the discrete gap probabilities in a multicritical random partitions model. Studying standard Riemann-Hilbert problem for orthogonal polynomials on the unit circle related to these Toeplitz determinants, we introduced a new Lax Pair for the discrete version of Painlevé II hierarchy. This hierarchy is obtained by studying compatibility condition for the Lax Pair and defined as a recursive operator iterated N-th times (2N-th is the order of the N-th discrete equation of the hierarchy).This work is based on a work in progress with Sofia Tarricone (UCLouvain).