Abstract
We show that the zeros of the random power series with i.i.d. real Gaussian coefficients form a Pfaffian point process. We further show that the product moments for absolute values and signatures of the power series can also be expressed by Pfaffians.
Highlights
Zeros of Gaussian processes have attracted much attention for many years both from theoretical and practical points of view
The first significant contribution to this study was made by Paley and Wiener [16]. They computed the expectation of the number of zeros of analytic Gaussian processes on a strip in the complex plane, which are defined as Wiener integrals
Kac gave an explicit expression for the probability density function of real zeros of a random polynomial n fn(z) = akzk k=0 with i.i.d. real standard Gaussian coefficients {ak}nk=0 and obtains precise asymptotics of the numbers of real zeros as n → ∞ [9]
Summary
Zeros of Gaussian processes have attracted much attention for many years both from theoretical and practical points of view. We will give a direct proof by using Hammersley’s formula for correlation functions of zeros of Gaussian analytic functions and a PfaffianHafnian identity due to Ishikawa, Kawamuko, and Okada [8] This is a similar way to that which was taken in [15] to prove that the zeros of fC form a determinantal point process, and in the process of our calculus for real zero correlations, we obtain new Pfaffian formulas for a real Gaussian process. Tn < 1, both the moments of absolute values E[|f (t1)f (t2) · · · f (tn)|] and those of signatures E[sgn(f (t1)) · · · sgn(f (tn))] are given by Pfaffians We stress that it should be surprising because such combinatorial formulas cannot be expected for general centered Gaussian processes.
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