@Article{Cho2019,
author = "Cho, Ilwoo and Jorgensen, Palle E. T.",
title = "Deformation of semicircular and circular laws via p-adic number fields and sampling of primes",
journal = "Opuscula Math.",
year = "2019",
volume = "39",
number = "6",
pages = "773--813",
doi = "https://doi.org/10.7494/OpMath.2019.39.6.773",
url = "https://doi.org/10.7494/OpMath.2019.39.6.773",
abstract = "In this paper, we study semicircular elements and circular elements in a certain Banach \(*\)-probability space \((\mathfrak{LS},\tau ^{0})\) induced by analysis on the \(p\)-adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). In particular, by truncating the set \(\mathcal{P}\) of all primes for given suitable real numbers \(t\lt s\) in \(\mathbb{R}\), two different types of truncated linear functionals \(\tau_{t_{1}\lt t_{2}}\), and \(\tau_{t_{1}\lt t_{2}}^{+}\) are constructed on the Banach \(*\)-algebra \(\mathfrak{LS}\). We show how original free distributional data (with respect to \(\tau ^{0}\)) are distorted by the truncations on \(\mathcal{P}\) (with respect to \(\tau_{t\lt s}\), and \(\tau_{t\lt s}^{+}\)). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.",
keywords = "free probability, primes, \(p\)-adic number fields, Banach \(*\)-probability spaces, semicircular elements, circular elements, truncated linear functionals",
issn = "1232-9274"
}