Abstract
We study real-valued asymptotic solutions of the discrete Painlevé equation of second type (dPII) \(x_{n+1} + x_{n-1} = \frac{n \,x_n}{\nu (x_n^2 - 1)}, \quad n \in {\mathbb N}.\) In the case of \(n/\nu = O(1)\) , and as \(n\to\infty\) , the asymptotics is nonuniform. Near the point \(n= 2\nu\) , an inner transition layer occurs, which matches regular asymptotics to the left and to the right of this point. The matching procedure involves classical Painlevé II transcendents. The asymptotics are applied to discrete gap probabilities and random matrix theory.
DOI 10.1134/S1061920824030130
