The heat flow, GAF, and SL(2;R)

2023/04/13 by Brian K. Hall, Ching-Wei Ho, Hall, Brian +5 · 1 citation
Mathematics · #Flow (mathematics) #Function (biology) #Geometry #Geometry and complex manifolds #Heat flow #Heat kernel #Invariant (physics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Meromorphic and Entire Functions #Physics #Quadratic equation #Thermodynamics

paper · pdf · doi:10.48550/arxiv.2304.06665

openalex publication_date 2023/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

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;ℝ).

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