2024/12/08 by Yan V. Fyodorov, Fyodorov, Yan V., Boris A. Khoruzhenko +3 · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Advanced Statistical Methods and Models #Data Management and Algorithms
paper · pdf · doi:10.48550/arxiv.2412.06086
We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form φ(z) = ∑k≥ 0 ck zk given that |φ(0)|2=t and assuming coefficients ck to be independent standard complex normals. We obtain the full conditional distribution of Nq, the number of zeros of φ(z) within a disk of radius √(q) centred at the origin, and prove its asymptotic normality in the limit when q→ 1-, the limit that captures the entire zero set of φ(z). In the same limit we also develop precise estimates for conditional probabilities of moderate to large deviations from normality. Finally, we determine the asymptotic form of Pk(t;q)=Prob \ Nq= k | |φ(0)|2=t \ in the limit when k is kept fixed whilst q approaches 1. To leading order, the hole probability P0(t;q) does not depend on t for t>0 but yet is different from that of P0(t=0;q) and coincides with the hole probability for unconditioned hyperbolic GAF of the form ∑k≥ 0 √(k+1) ck zk. We also find that asymptotically as q → 1-, Pk(t;q)= et Pk(0;q) for every fixed k ≥ 1 with Pk(0;q)= Prob \ Nq =k-1 \.