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  "title": "Smooth random radial profiles for isotropic Kac polynomials",
  "shortTitle": "Smooth profiles that remain random",
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  "oneLine": "Heavy-tailed coefficients leave a smooth but genuinely random microscopic distribution of root radii, with contrasting limits as the tail index approaches zero and two.",
  "abstract": "For isotropic independent complex Kac coefficients with positive regularly varying moduli of index alpha between zero and two, the empirical measure of n log|z| converges in distribution to a random probability measure. Its conditional phase potential yields an almost surely smooth density, while every fixed-threshold root fraction has full distributional support on [0,1]. The mean agrees with the classical stable expected-count profile. At the two tail-index endpoints, the already formed profile approaches respectively a common uniform random root fraction with mass escaping to infinity, and the deterministic Gaussian profile. These are iterated limits, not joint limits in degree and tail index.",
  "datePublished": "2026-10-11",
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    "name": "Does a heavy-tailed Kac polynomial have a deterministic microscopic radial profile?",
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    "random polynomials",
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    {
      "citation": "Ibragimov and Zeitouni (1997), On roots of random polynomials: earlier stable expected-count profile.",
      "url": "https://doi.org/10.1090/S0002-9947-97-01766-2",
      "doi": "10.1090/S0002-9947-97-01766-2"
    },
    {
      "citation": "Feldheim (2013), Zeroes of Gaussian analytic functions with translation-invariant distribution: structural precedent for random limiting zero measures.",
      "url": "https://arxiv.org/abs/1105.3929v3",
      "doi": "10.48550/arXiv.1105.3929"
    },
    {
      "citation": "Iksanov, Kabluchko and Marynych (2016), Local universality for real roots of random trigonometric polynomials: stable analytic-function limit machinery.",
      "url": "https://arxiv.org/abs/1601.05740v2",
      "doi": "10.48550/arXiv.1601.05740"
    },
    {
      "citation": "Kabluchko and Zaporozhets (2013), Roots of random polynomials whose coefficients have logarithmic tails: a different heavy-tail regime.",
      "url": "https://arxiv.org/abs/1110.2585v2",
      "doi": "10.48550/arXiv.1110.2585"
    },
    {
      "citation": "Feng, Gao and Zhou, Limit theorems associated with the Pitman–Yor process: established endpoint behaviour of normalized weights.",
      "url": "https://arxiv.org/abs/1602.08382v1",
      "doi": "10.48550/arXiv.1602.08382"
    },
    {
      "citation": "Lundqvist (2015), An explicit calculation of the Ronkin function: finite phase-average geometry and regularity.",
      "url": "https://www.numdam.org/article/AFST_2015_6_24_2_227_0.pdf",
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