2003/08/11 by Pavel Bleher, Pavel M. Bleher, Bleher, Pavel M. +2
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60D05 #FOS: Physical sciences #Financial Risk and Volatility Modeling #Geometry and complex manifolds #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #math-ph #math.MP #msc:60D05
paper · pdf · doi:10.48550/arxiv.math-ph/0308014
29 pages
arxiv created 2003/08/11 · openalex publication_date 2003/08/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The existence of the scaling limit and its universality, for correlations between zeros of \it Gaussian random polynomials, or more generally, \it Gaussian random sections of powers of a line bundle over a compact manifold has been proved in a great generality in the works [BBL2], [Ha], [BD], [BSZ1]-[BSZ4], and others. In the present work we prove the existence of the scaling limit for a class of \it non-Gaussian random polynomials. Our main result is that away from the origin the scaling limit exists and is universal, so that it does not depend on the distribution of the coefficients. At the origin the scaling limit is not universal, and we find a crossover from the nonuniversal asymptotics of the density of the probability distribution of zeros at the origin to the universal one away from the origin.