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

Brian Hall, Ching-Wei Ho, Jonas Jalowy, Zakhar Kabluchko

TL;DR

The paper analyzes how the heat-flow operator acts on entire functions of order at most 2, centering on the plane Gaussian analytic function (GAF). It proves that applying heat flow to a GAF, followed by a Gaussian dilation, yields another GAF with zeros distributed the same as the original up to a deterministic scaling, and it characterizes the evolution of individual zeros via a Calogero–Moser–type framework. The authors establish unitary metaplectic representations tying the heat flow to SL(2,ℝ) actions and prove hyperbolic-invariance properties of the zero set. They also develop a rigorous differential-analytic apparatus for the zeros under heat flow, including approximation by polynomials and explicit formulas in the order-2, finite-type regime, paving the way for a detailed dynamical understanding of zeros under Gaussian-analytic heat dynamics.

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 $SL(2;\mathbb{R}).$

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

TL;DR

The paper analyzes how the heat-flow operator acts on entire functions of order at most 2, centering on the plane Gaussian analytic function (GAF). It proves that applying heat flow to a GAF, followed by a Gaussian dilation, yields another GAF with zeros distributed the same as the original up to a deterministic scaling, and it characterizes the evolution of individual zeros via a Calogero–Moser–type framework. The authors establish unitary metaplectic representations tying the heat flow to SL(2,ℝ) actions and prove hyperbolic-invariance properties of the zero set. They also develop a rigorous differential-analytic apparatus for the zeros under heat flow, including approximation by polynomials and explicit formulas in the order-2, finite-type regime, paving the way for a detailed dynamical understanding of zeros under Gaussian-analytic heat dynamics.

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
Paper Structure (26 sections, 33 theorems, 217 equations, 2 figures)

This paper contains 26 sections, 33 theorems, 217 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Heat flow on entire functions
  3. Heat flow on the Gaussian analytic function
  4. Connection to the group $SL(2;\mathbb{R})$
  5. Heat flow on entire functions of order at most 2
  6. Order and type of a holomorphic function
  7. Definition of the heat flow
  8. Properties of the heat flow
  9. Evolution of the zeros
  10. Examples
  11. Gaussian analytic function undergoing the heat flow
  12. The main result
  13. Approximation by polynomials
  14. The evolution of individual zeros
  15. Connection to the group $SL(2;\mathbb{R})$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $G$ be the plane GAF and let $\tau$ be a complex number with $\left\vert \tau\right\vert <1.$ Then the random holomorphic function $V_{\tau}G$ given by is well defined and has the same distribution as $G.$ In particular, we have equality in distribution of the collections of zeros:

Figures (2)

  • Figure 1: Evolution of the 100 smallest zeros of the GAF under the heat flow for real times $\tau$ with $0\leq\tau<1$ (blue). The straight-line approximation $a+\tau\bar{a}$ to each curve is shown in gray. The curves start in the disk of radius 10 and come close to the $x$-axis as $\tau$ approaches 1.
  • Figure 2: Plots of the curves $z_{j}(t)-\tau\overline{z_{j}(0)},$ where $z_{j}(0)$ is a zero of the GAF, for $0\leq\tau<1.$ There is a small dot at the starting point $z_{j}(0)$ of each curve.

Theorems & Definitions (76)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Definition 2.4
  • ...and 66 more