Opuscula Math. 39, no. 6 (2019), 773-813

Opuscula Mathematica

Deformation of semicircular and circular laws via p-adic number fields and sampling of primes

Ilwoo Cho
Palle E. T. Jorgensen

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.

Mathematics Subject Classification: 11R56, 46L54, 47L30, 47L55.

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  • Ilwoo Cho
  • Saint Ambrose University, Department of Mathematics and Statistics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
  • Communicated by Petru A. Cojuhari.
  • Received: 2019-08-05.
  • Accepted: 2019-08-20.
  • Published online: 2019-11-22.
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Cite this article as:
Ilwoo Cho, Palle E. T. Jorgensen, Deformation of semicircular and circular laws via p-adic number fields and sampling of primes, Opuscula Math. 39, no. 6 (2019), 773-813, https://doi.org/10.7494/OpMath.2019.39.6.773

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