The image of random analytic functions: coverage of the complex plane via branching processes
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909209743851520 |
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| author | Nishry, Alon Paquette, Elliot |
| author_facet | Nishry, Alon Paquette, Elliot |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19380 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The image of random analytic functions: coverage of the complex plane via branching processes Nishry, Alon Paquette, Elliot Probability Complex Variables 60J85, 30D35, 30B20 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. |
| title | The image of random analytic functions: coverage of the complex plane via branching processes |
| topic | Probability Complex Variables 60J85, 30D35, 30B20 |
| url | https://arxiv.org/abs/2403.19380 |