Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation

TL;DR

This work develops a unified, probabilistic framework to determine the limiting zero distribution of polynomial sequences with nonpositive zeros via an exponential profile of their coefficients. By representing polynomials as generating functions of sums of independent Bernoulli variables and applying large deviation techniques, it proves an equivalence: a convergent empirical zero distribution is exactly characterized by the existence of an exponential profile, and the limiting measure is recoverable from, and determines, the profile. The paper then harnesses this profile approach to study how zeros behave under Hadamard products and finite free convolutions, and under repeated differentiation, providing new proofs and extending convergence results to noncompactly supported limits. It further develops free-probability tools (R- and S-transforms) in this profile context to derive additive and multiplicative convolutions for measures with atoms at infinity, and offers complete proofs for the repeated-differentiation regime, including detailed asymptotics and explicit transforms. Overall, the results unify and extend prior analytic and combinatorial approaches, offering a flexible, probabilistic pathway to zero-distribution problems and their polynomial operations.

Abstract

Given a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of $P_n$ as $\mathrm{deg}\, P_n \to \infty$. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution $μ$ and the existence of an exponential profile $g$ associated with the coefficients of the polynomials $(P_n)_{n \in \mathbb{N}}$. The exponential profile $g$, which can be roughly described by $[z^k]P_n(z) \approx \exp(n g(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $μ$: the functions $t \mapsto tG(t)$ and $α\mapsto \exp(-g'(α))$ are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. In particular, our approach yields new proofs of the convergence results `$\boxplus_n \to \boxplus$' and `$\boxtimes_n \to \boxtimes$', extending them to cases where the distributions are not necessarily compactly supported.

Figures (2)

• Figure 1: Roots of repeated derivatives of $p_n(z) = \prod_{j=0}^{n-1} \left(z - \frac{j}{n}\right)$ with $n=50$. The black dots have coordinates $(\frac{\ell}{n}, y_{i,\ell})$, where $\ell\in \{0,1,\ldots, n\}$ is the order of differentiation and $y_{1,\ell},\ldots, y_{n-\ell, \ell}\in [0,1]$ are the roots of $p_n^{(\ell)}$. The red curves show the support of the probability measure ${\rm Unif}_{[0,1]} \boxtimes {\nu}_{0,1;\kappa}$ as a function of $\kappa\in (0,1)$.
• Figure 2: Graphs of the functions $t\mapsto Y_{\kappa}(t)=\coth(t)-\kappa/t$ for $\kappa=0.1,0.3,0.5,0.7,0.9$.

Theorems & Definitions (58)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Theorem 2.2: Profile for coefficients implies limiting distribution of zeros
• Remark 2.3
• Theorem 2.4: Distribution of zeros implies exponential profile
• Remark 2.5
• Remark 2.6: A non-rigorous argument
• Proposition 2.7: Characterization of profiles
• Theorem 3.1