Opuscula Math. 39, no. 6 (2019), 773-813
https://doi.org/10.7494/OpMath.2019.39.6.773

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.
• Accepted: 2019-08-20.
• Published online: 2019-11-22.