{"date_updated":"2025-04-23T14:38:04Z","status":"public","department":[{"_id":"94"}],"abstract":[{"text":"Given a sequence of polynomials $(P_n)_{n \\in \\mathbb{N}}$ with only\r\nnonpositive zeros, the aim of this article is to present a user-friendly\r\napproach for determining the limiting zero distribution of $P_n$ as\r\n$\\mathrm{deg}\\, P_n \\to \\infty$. The method is based on establishing an\r\nequivalence between the existence of a limiting empirical zero distribution\r\n$\\mu$ and the existence of an exponential profile $g$ associated with the\r\ncoefficients of the polynomials $(P_n)_{n \\in \\mathbb{N}}$. The exponential\r\nprofile $g$, which can be roughly described by $[z^k]P_n(z) \\approx \\exp(n\r\ng(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $\\mu$:\r\nthe functions $t \\mapsto tG(t)$ and $\\alpha \\mapsto \\exp(-g'(\\alpha))$ are\r\nmutual inverses. This relationship, in various forms, has previously appeared\r\nin the literature, most notably in the paper [Van Assche, Fano and Ortolani,\r\nSIAM J. Math. Anal., 1987].\r\n  As a first contribution, we present a self-contained probabilistic proof of\r\nthis equivalence by representing the polynomials as generating functions of\r\nsums of independent Bernoulli random variables. This probabilistic framework\r\nnaturally lends itself to tools from large deviation theory, such as the\r\nexponential change of measure. The resulting theorems generalize and unify a\r\nrange of previously known results, which were traditionally established through\r\nanalytic or combinatorial methods.\r\n  Secondly, using the profile-based approach, we investigate how the\r\nexponential profile and the limiting zero distribution behave under certain\r\noperations on polynomials, including finite free convolutions, Hadamard\r\nproducts, and repeated differentiation. In particular, our approach yields new\r\nproofs of the convergence results `$\\boxplus_n \\to \\boxplus$' and `$\\boxtimes_n\r\n\\to \\boxtimes$', extending them to cases where the distributions are not\r\nnecessarily compactly supported.","lang":"eng"}],"_id":"59664","language":[{"iso":"eng"}],"publication":"arXiv:2504.11593","author":[{"id":"113768","full_name":"Jalowy, Jonas","last_name":"Jalowy","first_name":"Jonas","orcid":"0000-0001-9624-2685"},{"first_name":"Zakhar","last_name":"Kabluchko","full_name":"Kabluchko, Zakhar"},{"full_name":"Marynych, Alexander","last_name":"Marynych","first_name":"Alexander"}],"user_id":"113768","date_created":"2025-04-23T14:37:41Z","title":"Zeros and exponential profiles of polynomials I: Limit distributions,  finite free convolutions and repeated differentiation","external_id":{"arxiv":["2504.11593"]},"citation":{"ieee":"J. Jalowy, Z. Kabluchko, and A. Marynych, “Zeros and exponential profiles of polynomials I: Limit distributions,  finite free convolutions and repeated differentiation,” <i>arXiv:2504.11593</i>. 2025.","bibtex":"@article{Jalowy_Kabluchko_Marynych_2025, title={Zeros and exponential profiles of polynomials I: Limit distributions,  finite free convolutions and repeated differentiation}, journal={arXiv:2504.11593}, author={Jalowy, Jonas and Kabluchko, Zakhar and Marynych, Alexander}, year={2025} }","ama":"Jalowy J, Kabluchko Z, Marynych A. Zeros and exponential profiles of polynomials I: Limit distributions,  finite free convolutions and repeated differentiation. <i>arXiv:250411593</i>. Published online 2025.","mla":"Jalowy, Jonas, et al. “Zeros and Exponential Profiles of Polynomials I: Limit Distributions,  Finite Free Convolutions and Repeated Differentiation.” <i>ArXiv:2504.11593</i>, 2025.","chicago":"Jalowy, Jonas, Zakhar Kabluchko, and Alexander Marynych. “Zeros and Exponential Profiles of Polynomials I: Limit Distributions,  Finite Free Convolutions and Repeated Differentiation.” <i>ArXiv:2504.11593</i>, 2025.","apa":"Jalowy, J., Kabluchko, Z., &#38; Marynych, A. (2025). Zeros and exponential profiles of polynomials I: Limit distributions,  finite free convolutions and repeated differentiation. In <i>arXiv:2504.11593</i>.","short":"J. Jalowy, Z. Kabluchko, A. Marynych, ArXiv:2504.11593 (2025)."},"type":"preprint","year":"2025"}