Repeated differentiation and free unitary Poisson process

TL;DR

This paper derives a hydrodynamic limit for zeros of trigonometric polynomials under repeated differentiation, showing that the limit distribution equals the free multiplicative convolution of the initial zeros with the free unitary Poisson Π_t. The authors connect trig differentiation to finite free probability via a Ψ-map and circular Laguerre polynomials, and establish asymptotics through a saddle-point analysis of circular Laguerre polynomials and a principal branch ζ_t solving ζ − t tan ζ = θ. They provide explicit density and moment formulas for Π_t and link the results to Steinerberger’s PDE, offering a formal Fourier-based solution that describes the evolution of the zero distribution on the unit circle. Overall, the work integrates finite/free probability, complex analysis, and asymptotic techniques to describe the precise limiting behavior of zeros under iterative differentiation.

Abstract

We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree $d$ trigonometric polynomial are distributed according to some probability measure $ν$ in the large $d$ limit, then the zeroes of its $[2td]$-th derivative, where $t>0$ is fixed, are distributed according to the free multiplicative convolution of $ν$ and the free unitary Poisson distribution with parameter $t$. In the simplest special case, our result states that the zeroes of the $[2td]$-th derivative of the trigonometric polynomial $(\sin \frac θ2)^{2d}$ (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter $t$, in the large $d$ limit. The latter distribution is defined in terms of the function $ζ=ζ_t(θ)$ which solves the implicit equation $ζ- t \tan ζ= θ$ and satisfies $$ ζ_t(θ)= θ+ t \tan (θ+ t \tan (θ+ t \tan (θ+\ldots))), \qquad \mathrm{Im}\, θ>0, \;\; t>0. $$

Figures (6)

• Figure 1: Zeroes of $L_{n,k}(z)$ with $n=400$ and $k=[tn]$, where $t\in \{\frac{1}{7}, \frac{3}{7}, \frac{5}{7}, 1\}$. The multiplicity of the zero at $z=1$ is $n-k$.
• Figure 2: Level lines of $z\mapsto |f_s(z;\theta)|$ for $s=1$ and $\theta = 1 + 4{\rm{i}}$. A saddle-point contour is shown in black.
• Figure 3: The first two panels show the graphs of the functions $y\mapsto y - t \tanh y$, $y\geq 0$. Left: $0\leq t \leq 1$. Middle: $t>1$. The right panel shows the graphs of the functions $\tilde{y} \mapsto \tilde{y} - t \mathop{\mathrm{cotanh}}\limits \tilde{y}$, $\tilde{y}\geq 0$, for $0\leq t \leq 5$.
• Figure 4: The graph of the function $\operatorname{Im} \zeta_t(\theta)$ for real $\theta$. Left: $t=0.3$. Middle: $t=1$. Right: $t=1.5$.
• Figure 5: The graph of the function $\zeta\mapsto \zeta - t \tan \zeta$, $\zeta\in \mathbb{R}$. Left: $t=0.3$. Middle: $t=1$. Right: $t=1.5$.

Theorems & Definitions (67)

• Conjecture 1.1: cf. steinerberger_realorourke_steinerberger_nonlocalbogvad_etal
• Theorem 1.2
• Example 1.3: The "fundamental solution"
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof