TY  - JOUR
ID  - Cho2019
LB  - Cho2019
AU  - Cho, Ilwoo
AU  - Jorgensen, Palle E. T.
TI  - Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
ST  - Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2019
VL  - 39
IS  - 6
SP  - 773
EP  - 813
DO  - https://doi.org/10.7494/OpMath.2019.39.6.773
UR  - https://doi.org/10.7494/OpMath.2019.39.6.773
AB  - 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.
KW  - free probability
KW  - primes
KW  - \(p\)-adic number fields
KW  - Banach \(*\)-probability spaces
KW  - semicircular elements
KW  - circular elements
KW  - truncated linear functionals
SN  - 1232-9274
ER  -