Pavel Wiegman: Deformation of Selberg-Dyson Integral and Aspects of Complex Geometry

Abstract.

The talk is based on a (not so) recent paper with Anton Zabrodin, where we discussed an ensemble of particles with logarithmic repulsive interaction on a Jordan curve $\Gamma$. Such a problem is described by a geometric deformation of the Dyson-Selberg integral

\(\oint_{\Gamma}\prod_{I<j} |z_i-z_j|^{2\beta} dz_1\cdots dz_n\)

In the limit of a large number of variables, the integral converges to the spectral determinant of the Neumann jump operator defined on the curve. Equivalently, it converges to the Fredholm determinant of the Neumann–Poincare operator. These results suggest that the Dyson-Selberg integral utilizes an extension of Fekete's theory which deals with the finite-dimensional approximation of conformal maps.