Event
Ferenc Balogh, John Abbott College et CRM
Tuesday, March 6, 2018 15:30to16:30
Room 4336, Pav. Andr茅-Aisenstadt, 2920, ch. de la Tour, CA
On the distribution of poles of rational solutions to the Painleve II hierarchy
The second Painlev茅 equation admits a sequence of rational solutions that can be written as logarithmic derivatives of special evaluations of Schur polynomials associated to staircase-shaped integer partitions. These polynomials are known in the literature as Vorobev-Yablonsky polynomials, and their algebraic structure and asymptotic properties are fairly well understood. In particular, Bertola and Bothner found that the squares of these polynomials can be written as partition functions ((Hankel determinants) of a non-hermitian matrix model on a contour and therefore their asymptotic behaviour may be investigated by the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems. The limiting distribution of the poles of the rational solutions is encoded into the geometry of the steepest descent contours, and the boundary of the pole accumulation region can be characterized as the locus of points where the associated algebraic curve changes its genus. The second Painlev茅 equation is the first member of the Painlev茅 II hierarchy of integrable nonlinear ODEs. All members of the PII hierarchy admit a sequence of rational solutions, and they are all expressible in terms of special evaluations of Schur polynomials associated to staircase partitions. The matrix model representation of Bertola and Bothner generalizes naturally to all rational solutions of the PII hierarchy, and the Deift-Zhou nonlinear steepest descent method offers an insight into the geometry of their limiting pole structure. In this talk I will introduce Schur polynomials associated to staircase partitions and explain their interpretation as matrix model partition functions. I will show how the Riemann-Hilbert method helps to explain the curious limiting shapes of the pole accumulation regions that were observed originally by Clarkson and Mansfield via numerical methods. The talk is based on a joint work with M. Bertola and T. Bothner.