Alexander Marynych

Given a sequence of polynomials $$(P_n)_{n \in \mathbb {N}}$$ ( P n ) n ∈ N with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of $$P_n$$ P n as $$\deg P_n \rightarrow \infty $$ deg P n → ∞ . The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution $$\mu $$ μ and the existence of an exponential profile g associated with the coefficients of the polynomials $$(P_n)_{n \in \mathbb {N}}$$ ( P n ) n ∈ N . The exponential profile g , which can be roughly described by $$[z^k]P_n(z) \approx \exp \big (n g(k/n)\big )$$ [ z k ] P n ( z ) ≈ exp ( n g ( k / n ) ) , offers a direct route to computing the Cauchy transform G of $$\mu $$ μ : the functions $$t \mapsto tG(t)$$ t ↦ t G ( t ) and $$\alpha \mapsto \exp \big (-g'(\alpha )\big )$$ α ↦ 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 ‘ $$\boxplus _n \rightarrow \boxplus $$ ⊞ n → ⊞ ’ and ‘ $$\boxtimes _n \rightarrow \boxtimes $$ ⊠ n → ⊠ ’, extending them to cases where the distributions are not necessarily compactly supported. This paper is the first in a series of three. In Part II [36], we explore a wide range of examples, covering many classical families of polynomials, while Part III [37] will focus on random polynomials and establish functional limit theorems for their random profiles.

In [Jalowy, Kabluchko, Marynych, Part I of the present work, Constr. Approx.], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $$\infty $$ ∞ . In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little q -Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.

For a right-continuous nondecreasing and unbounded function $V$ of at most exponential growth, which vanishes on the negative halfline, we investigate the asymptotic behavior of the Lebesgue-Stieltjes convolution powers $V^{\ast(j)}(t)$ as both $j$ and $t$ tend to infinity. We obtain a comprehensive asymptotic formula for $V^{\ast(j)}(t)$, which is valid across different regimes of simultaneous growth of $j$ and $t$. Our main technical tool is an exponential change of measure, which is a standard technique in the large deviations theory. Various applications of our result are given.

Let $({\xi _{1}},{\eta _{1}})$, $({\xi _{2}},{\eta _{2}}),\dots $ be independent identically distributed ${\mathbb{N}^{2}}$-valued random vectors with arbitrarily dependent components. The sequence ${({\Theta _{k}})_{k\in \mathbb{N}}}$ defined by ${\Theta _{k}}={\Pi _{k-1}}\cdot {\eta _{k}}$, where ${\Pi _{0}}=1$ and ${\Pi _{k}}={\xi _{1}}\cdot \dots \cdot {\xi _{k}}$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets $\{{\Pi _{1}},{\Pi _{2}},\dots ,{\Pi _{k}}\}\subset \mathbb{N}$ and $\{{\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{k}}\}\subset \mathbb{N}$, $k\in \mathbb{N}$, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.