Abstract
In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is “fully rigid”. This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.
Highlights
Zero sets of random analytic functions, and especially Gaussian ones, have attracted the attention of researchers from various areas of mathematics in the last two decades.Given a sequence {an}n≥0 of non-negative numbers, we consider the random Taylor series f (z) = ξnanzn, (1.1)n≥0 where {ξn}n≥0 is a sequence of independent standard complex Gaussians
We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function
Denote by Zf = f −1 {0} the zero set of f ; its properties are determined by the covariance kernel
Summary
Zero sets of random analytic functions, and especially Gaussian ones, have attracted the attention of researchers from various areas of mathematics in the last two decades. One model that was well studied is the Gaussian Entire Function (GEF), given by the Taylor series. To study the rigidity of the zero set of a Gaussian entire function we impose certain conditions on its covariance kernel. We relate the rigidity of the zero set, to a certain growth condition on the function G, see the statement of Theorem 1.2. Two examples where the zero set exhibits full rigidity, include the kernel functions zn G(z) = exp (exp (z)) , and G(z) = logn (n + e) , n≥0 see Section 3.2. Another approach to full rigidity can be found in [2]. 1.1 Admissible kernel functions Given an entire function G with non-negative Taylor coefficients, put. The class of admissible functions has many nice closure properties, for example if f is admissible, so is ef , for details and examples, see [5]
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