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Mathematics > Probability

arXiv:1811.12225 (math)
[Submitted on 29 Nov 2018 (v1), last revised 6 Apr 2020 (this version, v2)]

Title:Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background

Authors:Raphael Butez, David García-Zelada
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Abstract:We study the outliers for two models which have an interesting connection. On the one hand, we study a specific class of planar Coulomb gases which are determinantal. It corresponds to the case where the confining potential is the logarithmic potential of a radial probability measure. On the other hand, we study the zeros of random polynomials that appear to be closely related to the first model. Their behavior far from the origin is shown to depend only on the decaying properties of the probability measure generating the potential. A similar feature is observed for their behavior near the origin. Furthermore, in some cases, the appearance of outliers is observed, and the zeros of random polynomials and the Coulomb gases are seen to exhibit exactly the same behavior, which is related to the unweighted Bergman kernel.
Comments: 36 pages, 4 figures
Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
MSC classes: Primary: 60G55, Secondary: 82B21, 60F05, 60K35, 30C15
Cite as: arXiv:1811.12225 [math.PR]
  (or arXiv:1811.12225v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1811.12225
Journal reference: Annals of Applied Probability 32 (2022), Issue 1, 392-425

Submission history

From: David García Zelada [view email]
[v1] Thu, 29 Nov 2018 14:59:18 UTC (47 KB)
[v2] Mon, 6 Apr 2020 19:54:15 UTC (311 KB)
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