2016/08/31 by Subhroshekhar Ghosh, Ghosh, Subhroshekhar, Alon Nishry +1 · 1 citation
Mathematics · #Geometry and complex manifolds #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1609.00084
We consider the Gaussian Entire Function (GEF) whose Taylor coefficients are\nindependent complex-valued Gaussian variables, and the variance of the kth\ncoefficient is 1/k!. This random Taylor series is distinguished by the\ninvariance of its zero set with respect to the isometries of the complex plane.\n We show that the law of the zero set, conditioned on the GEF having no zeros\nin a disk of radius r, and properly normalized, converges to an explicit\nlimiting Radon measure in the plane, as r goes to infinity. A remarkable\nfeature of this limiting measure is the existence of a large 'forbidden region'\nbetween a singular part supported on the boundary of the (scaled) hole and the\nequilibrium measure far from the hole.\n