Zero distribution of multiplicative Hermite and Laguerre polynomials
Zakhar Kabluchko
TL;DR
The paper introduces multiplicative analogues of Hermite and Laguerre polynomials, $H_n^*(x;s)$ and $L_n^*(x;b,c)$, and proves that their zeros, under natural degree-time scalings, converge to free multiplicative normal and free multiplicative Poisson distributions, respectively, on the positive half-line or the unit circle. A unified method based on a multiplicative heat flow and Burgers-type PDE for generating functions (and, in the Laguerre case, a discrete-time difference framework with Bell polynomials) yields the limiting analytic moments and their associated $S$-transforms: $S(z;s)=e^{-s(z+1/2)}$ for Hermite and $S(y)=\exp\left(\frac{\gamma}{\beta+1+y}\right)$ for Laguerre. The work provides a rigorous, self-contained route from finite-$n$ polynomial evolutions to free-probability limits, including explicit moment formulas and the structure of the limiting measures $\mu^{(s)}$ and $\nu_{\beta,\gamma}$. By bridging classical polynomial zero-distribution results with free convolution theory, the paper broadens the landscape of asymptotic spectral questions to multiplicative, unitary, and positive-domain settings, with potential implications for random matrix products and related stochastic processes.
Abstract
It is well-known that, as $n\to\infty$, the zero distribution of the $n$-th Hermite polynomial converges to the semicircular law (the free normal distribution), while the zero distribution of the associated Laguerre polynomials converges to the Marchenko--Pastur law (the free Poisson distribution). In this paper, we establish multiplicative analogues of these results. We define the multiplicative Hermite and Laguerre polynomials by \begin{align*} H_n^*(x;s) &:= e^{-\frac 12 s ((x\partial_x)^2 - n x \partial_x) } (x-1)^n = \sum_{j=0}^n (-1)^{n-j} \binom nj e^{-\frac 12 s (j^2 - nj)} x^j, \\ L_n^*(x; b,c) &:= (x\partial_x + b)^c (x-1)^n = \sum_{j=0}^n (-1)^{n-j} \binom nj (j+b)^c x^j, \end{align*} where $n\in \mathbb N_0$, $\partial_x$ denotes the differentiation operator w.r.t. $x$, and $s\in \mathbb R$, $b\in \mathbb C$, $c\in \mathbb N_0$ are parameters. In the Hermite case, we show that, as $n\to\infty$, the zero distribution of $H_n^*(x;s/n)$ converges weakly to the free multiplicative normal distribution on the positive half-line (when $s>0$) or to the free unitary normal distribution on the unit circle $\{|z| = 1\}$ (when $s<0$). In the Laguerre case, we show that the zero distribution of $L_n^*(x; nβ, \lfloor n γ\rfloor)$ converges to the free multiplicative Poisson distribution on the positive half-line (when $γ>0$ and $β\in \mathbb R\backslash[0,1]$) or on the unit circle (when $γ>0$ and $β\in -\frac 12 + \sqrt{-1} \, \mathbb R$). All these results are obtained by essentially the same method, which treats the Hermite/Laguerre cases and the unitary/positive settings in a unified way.