Real zeros of random analytic functions associated with geometries of\n constant curvature

2018/02/07 by Hendrik Flasche, Flasche, Hendrik, Zakhar Kabluchko +1
Mathematics · Social Sciences · #26C10 (Primary) #30C15 #60F05 #60F17 #60F99 #60G15 (Secondary) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Historical Geography and Cartography #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1802.02390

openalex publication_date 2018/02/07 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let \ξ₀, \ξ₁, \… be i.i.d. random variables with zero mean and unit\nvariance. We study the following four families of random analytic functions:\n\∑_k=0ⁿ \√( binom nk) \ξ_k z^k (spherical polynomials),\n\∑_k=0^\∞ \√(\(n^k)/(k!)) \ξ_k z^k (flat random analytic\nfunction), \∑_k=0^\∞ \√ binom n+k-1 k \ξ_k z^k (hyperbolic\nrandom analytic functions), \∑_k=0ⁿ \√(\(n^k)/(k!)) \ξ_k z^k\n(Weyl polynomials). We compute explicitly the limiting mean density of real\nzeroes of these random functions. More precisely, we provide a formula for\n\lim_n\→\∞ n^-1/2 \𝔼N_n[a,b], where N_n[a, b] is the\nnumber of zeroes in the interval [a,b].\n

Citations

• Expected number of real zeros of random Taylor Series

Related

• Real Zeros of Random Sums with I.I.D. Coefficients
• The density of complex zeros of random sums
• Zeros of Random Analytic Functions
• Number variance of random zeros
• Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series