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Zeros of polynomial powers under the heat flow

Antonia Höfert, Jonas Jalowy, Zakhar Kabluchko

TL;DR

The paper analyzes how zeros of high-degree polynomial powers evolve under the holomorphic backward heat flow. By representing P_t^n(z) as a contour integral and applying a refined saddle point analysis, it proves the existence of a limiting zero distribution μ_t on ℂ, describes μ_t via a self-consistent Stieltjes transform m_t and a maximally relevant saddle u_t^*(z), and characterizes μ_t as supported on finitely many smooth curves with densities determined by saddle-point transitions. It uncovers rich time asymptotics: small times yield local semicircular growth near initial zeros, large times yield a horizontal-line attractor whose macroscopic rescaling converges to the semicircle law, and it establishes a Hamilton–Jacobi/Burgers PDE structure for the logarithmic potential and Stieltjes transform. A simple d=1 example recovers the shifted semicircle, illustrating the theory concretely and connecting to free-probability and potential-theoretic viewpoints.

Abstract

We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $μ_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $μ_t$ is available.

Zeros of polynomial powers under the heat flow

TL;DR

The paper analyzes how zeros of high-degree polynomial powers evolve under the holomorphic backward heat flow. By representing P_t^n(z) as a contour integral and applying a refined saddle point analysis, it proves the existence of a limiting zero distribution μ_t on ℂ, describes μ_t via a self-consistent Stieltjes transform m_t and a maximally relevant saddle u_t^*(z), and characterizes μ_t as supported on finitely many smooth curves with densities determined by saddle-point transitions. It uncovers rich time asymptotics: small times yield local semicircular growth near initial zeros, large times yield a horizontal-line attractor whose macroscopic rescaling converges to the semicircle law, and it establishes a Hamilton–Jacobi/Burgers PDE structure for the logarithmic potential and Stieltjes transform. A simple d=1 example recovers the shifted semicircle, illustrating the theory concretely and connecting to free-probability and potential-theoretic viewpoints.

Abstract

We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial , we prove that the empirical zero distribution of its heat-evolved -th power converges to a distribution on the complex plane as tends to infinity. We describe this limit distribution as a function of the time parameter of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for is available.

Paper Structure

This paper contains 16 sections, 23 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Existence of a limiting distribution
  4. Saddle point method
  5. Analysis of the saddle point equation
  6. Limiting logarithmic potential
  7. Choice of Contour
  8. Application of the saddle point method
  9. Limit of the Empirical Root Measure
  10. Description of the limiting distribution
  11. Proof of Theorem \ref{['thm:description']}
  12. Large $t$ asymptotics
  13. Proof of Theorem \ref{['thm:semicircle']}
  14. Small $t$ asymptotics
  15. Proof of Theorem \ref{['thm:PDEs']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

For each fixed $t\in\mathbb{C}$, the empirical zero distribution $\mu_{n,t}$ of $P_t^n$ converges weakly to some compactly supported probability measure $\mu_t$ on the complex plane as $n\to\infty$. The limiting measure is singular with respect to the Lebesgue measure on $\mathbb{C}$. If $t=|t|e^{i\

Figures (3)

  • Figure 1: Evolution of heat-evolved polynomial powers $P_t^n$ for $d=6$ different zeros, $\lambda_1=-i$ with multiplicity $\alpha_1=5$ and five simple zeros $\lambda_2,\dots\lambda_6$ somewhere in the upper half-plane. The $\alpha n=200$ zeros are depicted as black dots with their trajectories in orange. For small time $t=1/5$ (left), we see small semicircle components of $\mu_t$ emerging from the respective initial zeros $\lambda_k$. At arbitrary time $t\approx 1.8$ (center) these lines become intricate to describe analytically and start to merge like a "zipper". For large time $t=10$ (right) only one curve remains in the support of $\mu_t$, passing through the center of mass (here, $=0$), which unfolds into the semicircular law when zooming out.
  • Figure 2: The curves in $\mathcal{D}_t^c$ in blue for $d=2$ with $\lambda_1=i,\lambda_2=-i$ and $t=2$, $t=4$ and $t=8$, including the finite $n=60$ approximation of $\mu_t$ of individual zeros (black), their trajectories (in orange) and the branching points (red).
  • Figure 3: The function $u\mapsto G(z,u)$ for $z=0$ and $d=2$ initial roots at $\lambda_1=1$ and $\lambda_2=-i$ after times $t=1/2$ (left) and $t=3$ (right). The three saddle points are depicted as black dots and the level height $\Sigma (h)$ at $h=G(0,u_t^*(0))$ of the maximally relevant saddle point is the orange area where our contour will lie. For smaller $t=1/2$, the behavior of $G$ is dominated by its component $\operatorname{Re}(u^2)$. The maximally relevant saddle point $u_t^*(0)\approx (1+i)/10$ is near $z=0$, we have an irrelevant saddle point near $\lambda_1$, and a saddle point of lower height than the maximally relevant one near $\lambda_2$. This illustrates our findings of Lemma \ref{['lem:small_t']} (in particular \ref{['eq:sp_dist']}) and Lemma \ref{['lem:u_t_small_t']}. For larger $t=3$, the function $G$ flattens out. No saddle point is irrelevant and the maximally relevant one at $u_t^*(0)\approx (3+8i)/10$ will eventually approach $u^+_t(0)=i\sqrt t$ as $t$ grows, another one moving towards $\lambda_1'=(1-i)/2$, and the last towards $u_t^-(0)=-i\sqrt t$. We will verify this in Lemma \ref{['lem:large_t']} and Corollary \ref{['cor:large_t_rescaled']} below.

Theorems & Definitions (59)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 49 more