The heat flow, GAF, and SL(2;R)
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917122634940416 |
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| author | Hall, Brian Ho, Ching-Wei Jalowy, Jonas Kabluchko, Zakhar |
| author_facet | Hall, Brian Ho, Ching-Wei Jalowy, Jonas Kabluchko, Zakhar |
| contents | We establish basic properties of the heat flow on entire holomorphic functions that have order at most 2. We then look specifically at the action of the heat flow on the Gaussian analytic function (GAF). We show that applying the heat flow to a GAF and then rescaling and multiplying by an exponential of a quadratic function gives another GAF. It follows that the zeros of the GAF are invariant in distribution under the heat flow, up to a simple rescaling.
We then show that the zeros of the GAF evolve under the heat flow approximately along straight lines, with an error whose distribution is independent of the starting point. Finally, we connect the heat flow on the GAF to the metaplectic representation of the double cover of the group $SL(2;\mathbb{R}).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06665 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The heat flow, GAF, and SL(2;R) Hall, Brian Ho, Ching-Wei Jalowy, Jonas Kabluchko, Zakhar Probability Classical Analysis and ODEs Complex Variables Primary: 30C15, Secondary: 35K05, 60G15, 30D20, 30D10, 34A99, 30H20 We establish basic properties of the heat flow on entire holomorphic functions that have order at most 2. We then look specifically at the action of the heat flow on the Gaussian analytic function (GAF). We show that applying the heat flow to a GAF and then rescaling and multiplying by an exponential of a quadratic function gives another GAF. It follows that the zeros of the GAF are invariant in distribution under the heat flow, up to a simple rescaling. We then show that the zeros of the GAF evolve under the heat flow approximately along straight lines, with an error whose distribution is independent of the starting point. Finally, we connect the heat flow on the GAF to the metaplectic representation of the double cover of the group $SL(2;\mathbb{R}).$ |
| title | The heat flow, GAF, and SL(2;R) |
| topic | Probability Classical Analysis and ODEs Complex Variables Primary: 30C15, Secondary: 35K05, 60G15, 30D20, 30D10, 34A99, 30H20 |
| url | https://arxiv.org/abs/2304.06665 |