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The image of random analytic functions: coverage of the complex plane via branching processes
- Publication Year :
- 2024
-
Abstract
- We consider the range of random analytic functions with finite radius of convergence. We show that any unbounded random Taylor series with rotationally invariant coefficients has dense image in the plane. We moreover show that if in addition the coefficients are complex Gaussian with sufficiently regular variances, then the image is the whole complex plane. We do this by exploiting an approximate connection between the coverage problem and spatial branching processes. This answers a long-standing open question of J.-P. Kahane, with sufficient regularity.<br />Comment: Updated to remove unused lemma and changed formatting
- Subjects :
- Mathematics - Probability
Mathematics - Complex Variables
60J85, 30D35, 30B20
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2403.19380
- Document Type :
- Working Paper