Distribution of the first particle in discrete orthogonal polynomial ensembles

Alexei Borodin; Dmitriy Boyarchenko

Distribution of the first particle in discrete orthogonal polynomial ensembles

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arxiv math-ph/0204001 v1 pith:NXLQFXQC submitted 2002-04-01 math-ph cond-mathep-thmath.CAmath.MPnlin.SI

Alexei Borodin , Dmitriy Boyarchenko This is my paper · ORCID

classification math-phcond-mathep-thmath.CAmath.MPnlin.SI

keywords discretedifferencedistributionfirstfunctionorthogonalparticlepolynomial

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We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational. In a number of classical special cases the recurrence procedure is equivalent to the difference and q-Painleve equations of chao-dyn/9507010, [Sakai]. Our approach is based on the formalism of discrete integrable operators and discrete Riemann--Hilbert problems developed in math.CO/9912093, math-ph/0111008.

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Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlev\'e III$_3$ tau functions
math-ph 2025-01 conditional novelty 7.0 of 10

A Fredholm determinant is proved to be the tau function of q-Painlevé III3, with its expansion matching Nekrasov partition functions and new explicit connection formulas.

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