Specialization: 011200 Physics
Program: 07 Methods and problems of mathematical and computational physics
Head of the program: prof. T.A. Suslina
Department of higher mathematics and mathematical physics
Advisor: doc. V.V. Sukhanov
Reviewer: doc. A.A. Kapaev

Asymptotic analysis of orthogonal polynomials on the real line with a discontinuous Gaussian weight: the critical case.
A.K. Bogatskii

We study orthogonal polynomials on the real line with a weight that is the product of the Gaussian weight and a piecewise constant complex-valued function with one discontinuity. The point of the discontinuity depends on a real parameter and approaches the edge of the equilibrium measure as the number of the polynomial increases. We calculate the asymptotics of the recurrence coefficients in this limit. The resulting asymptotics include a special solution to the 34th Painlevé equation, whose asymptotics along the real line are also calculated in the present work. In particular, for some values of the parameter of the weight function, this solution has singular asymptotics.
As an application of the above mentioned results, we prove the critical asymptotics of the Hankel determinant with a discontinuous Gaussian symbol, which also includes the 34th Painlevé transcendent. Additionally, several probabilistic interpretations of the results are suggested.

The list of the publications
1.	A. Bogatskiy, T. Claeys, A. Its. Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge. (in preparation)
2.	A. Bogatskiy. Asymptotics of a special singular solution of the Painlevé XXXIV equation. (in preparation)