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Zeros of random polynomials undergoing the heat flow

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

TL;DR

We study how zeros of high-degree random polynomials evolve under the complex heat flow and connect this evolution to a heat-flow analogue of random-matrix universality. Using a variational/logarithmic-potential framework and an explicit transport map, we derive the limiting zero distributions ν_t for Weyl polynomials and for a broad class of rotationally invariant initial zeros, identifying explicit critical times t_sing and t_Wig and showing push-forward structure before singularities. The analysis links to PDEs (Hamilton–Jacobi and Burgers), optimal transport (Wasserstein geodesics), and free probability (Brown measures and free convolution), yielding a rigorous polynomial-analytic counterpart to heat-flow heuristics. The results are illustrated with explicit examples, including Weyl, Littlewood–Offord, random entire functions, and evenly spaced roots, and they illuminate how zeros migrate under diffusion and how phase transitions to real-line concentration occur.

Abstract

We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law, as was recently conjectured for characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover, for a general family of random polynomials with independent coefficients and isotropic limiting distribution of zeros, we determine the zero distribution of the heat-evolved polynomials in terms of its logarithmic potential. Furthermore, we explicitly identify two critical time thresholds, at which singularities develop and at which the limiting distribution collapses to the semicircle law. We completely characterize the limiting root distribution of the heat-evolved polynomials before singularities develop as the push-forward of the initial distribution under a transport map. Finally, we discuss the results from the perspectives of partial differential equations (in particular Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free probability. The theory is accompanied by explicit examples, simulations, and conjectures.

Zeros of random polynomials undergoing the heat flow

TL;DR

We study how zeros of high-degree random polynomials evolve under the complex heat flow and connect this evolution to a heat-flow analogue of random-matrix universality. Using a variational/logarithmic-potential framework and an explicit transport map, we derive the limiting zero distributions ν_t for Weyl polynomials and for a broad class of rotationally invariant initial zeros, identifying explicit critical times t_sing and t_Wig and showing push-forward structure before singularities. The analysis links to PDEs (Hamilton–Jacobi and Burgers), optimal transport (Wasserstein geodesics), and free probability (Brown measures and free convolution), yielding a rigorous polynomial-analytic counterpart to heat-flow heuristics. The results are illustrated with explicit examples, including Weyl, Littlewood–Offord, random entire functions, and evenly spaced roots, and they illuminate how zeros migrate under diffusion and how phase transitions to real-line concentration occur.

Abstract

We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law, as was recently conjectured for characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover, for a general family of random polynomials with independent coefficients and isotropic limiting distribution of zeros, we determine the zero distribution of the heat-evolved polynomials in terms of its logarithmic potential. Furthermore, we explicitly identify two critical time thresholds, at which singularities develop and at which the limiting distribution collapses to the semicircle law. We completely characterize the limiting root distribution of the heat-evolved polynomials before singularities develop as the push-forward of the initial distribution under a transport map. Finally, we discuss the results from the perspectives of partial differential equations (in particular Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free probability. The theory is accompanied by explicit examples, simulations, and conjectures.
Paper Structure (40 sections, 33 theorems, 227 equations, 10 figures)

This paper contains 40 sections, 33 theorems, 227 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Heat-flow operator and notation
  4. Weyl polynomials
  5. Polynomials with general rotationally invariant distributions of zeros
  6. Properties of the logarithmic potential
  7. The transport map and push-forward theorems
  8. Heuristics
  9. Definition and properties of the transport map
  10. The global push-forward theorem
  11. The local push-forward theorem and the density at regular points
  12. Examples
  13. Weyl polynomials
  14. Littlewood--Offord polynomials
  15. Random entire functions
  16. ...and 25 more sections

Key Result

Theorem 2.1

For every $t\in \mathbb{C}$, the empirical measure of zeros of the polynomial $\exp\{-\frac{t}{2n} \partial_z^2\} W_n$, viewed as a random element with values in $\mathcal{M}(\mathbb{C})$, converges in probability to a deterministic probability measure $\nu_t$ that can be described as follows.

Figures (10)

  • Figure 1: Zeros of $\exp\{-\frac{t}{2n} \partial_z^2\} W_n$ with $t=0$, $t={\rm{i}}/2$, $t=-99/100\cdot {\rm{i}}$. The degree is $n=1000$.
  • Figure 2: Samples of the empirical root distribution $\llbracket P_n(z;t)\rrbracket$ of heat-evolved random polynomials of degree $n=1000$ after time $t=1/2$. Here, the heat evolution started with Kac polynomials (left) from Example \ref{['ex:Kac']}, Weyl polynomials (center) from Example \ref{['ex:Weyl']} and Littlewood--Offord polynomials (right) from Example \ref{['ex:LO']}. Theorem \ref{['theo:main_general_g']} describes the limiting distributions. Animated versions of this figure (among others) can be found in "poly_heat_flow_animated.pdf" of the supplementary files on arxiv.org.
  • Figure 3: The logarithmic potential $\Psi$ of the semicircle law (left), the logarithmic potential $U_t$ of the elliptic law (center) for $t=1/2$, and its approximation (see Theorem \ref{['theo:main_weyl']}) for finite $n=20$, i.e. the logarithmic potential of $\llbracket P_{n}(z;t)\rrbracket$ (right). In the proof of Theorem \ref{['theo:main_general_g']} it will be crucial that the logarithmic potentials at finite $n$ approximate the limiting logarithmic potentials very well, except at the positions of their singularities.
  • Figure 4: Dynamics of the roots of the heat-evolved random polynomial $P_n(z;t)$ degree $n=300$ from $t=0$ until $t=1/2$. Again, we compare Kac polynomials (left) from Example \ref{['ex:Kac']}, Weyl polynomials (center) from Example \ref{['ex:Weyl']} and Littlewood--Offord polynomials (right) from Example \ref{['ex:LO']}. For Weyl polynomials, Theorem \ref{['theo:pushforward']} explains that the limiting distribution are push-forwards under $T_t$, which according point into the direction of the Stieltjes transform.
  • Figure 5: "Pairing" of the roots $z_j(t)$ (black dots) with the predicted locations of the transport map $T_t(z_j(0))$ (red dots) after time $t=2/3$. For $n=300$, we compare the heat evolution of a polynomial with i.i.d. roots (top left), the characteristic polynomial of Ginibre matrices (top right), Weyl polynomial (bottom left) and regular lattice in the unit disk (bottom right). The pairing appears to improve as the point process at time $0$ gets more organized. Note the curious line structures which appear for i.i.d. zeros are also barely visible for eigenvalues (top right), but not for Weyl polynomials.
  • ...and 5 more figures

Theorems & Definitions (82)

  • Theorem 2.1
  • Theorem 2.2: Kabluchko--Zaporozhets
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6: First Main Theorem
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 72 more