Zeros and Exponential Profiles of Polynomials I: Limit Distributions, Finite Free Convolutions and Repeated Differentiation

Abstract 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 exponent