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. Such a problem is described by a geometric deformation of the celebrated Dyson-Selberg integral
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. These results suggest that the Dyson-Selberg integral exhibits an emergent conformal covariance and utilize a probabilistic version of theFekete's theory of the finite-dimensional approximation of conformal maps.
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