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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  1. D. Alpay, P.E.T. Jorgensen, D. Levanony, A class of Gaussian processes with fractional spectral measures, J. Funct. Anal. 261 (2011) 2, 507-541.
  2. M. de Boeck, A. Evseev, S. Lyle, and L. Speyer, On bases of some simple modules of symmetric groups and Hecke algebras, Transform. Groups 23 (2018) 3, 631-669.
  3. I. Cho, Free distributional data of arithmetic functions and corresponding generating functions, Complex Anal. Oper. Theory 8 (2014) 2, 537-570.
  4. I. Cho, Dynamical systems on arithmetic functions determined by primes, Banach J. Math. Anal. 9 (2015) 1, 173-215.
  5. I. Cho, On dynamical systems induced by \(p\)-adic number fields, Opuscula Math. 35 (2015) 4, 445-484.
  6. I. Cho, Free semicircular families in free product Banach \(*\)-algebras induced by \(p\)-adic number fields over primes \(p\), Complex Anal. Oper. Theory 11 (2017) 3, 507-565.
  7. I. Cho, Asymptotic semicircular laws induced by \(p\)-adic number fields \(\mathbb{Q}_p\) over primes, Complex Anal. Oper. Theory 13 (2019) 7, 3169-3206.
  8. I. Cho, T.L. Gillespie, Free probability on the Hecke algebra, Complex Anal. Oper. Theory 9 (2015) 7, 1491-1531.
  9. I. Cho, P.E.T. Jorgensen, Krein-Space Operators Induced by Dirichlet Characters, Commutative and Noncommutative Harmonic Analysis and Applications, Contemp. Math., vol. 603, Amer. Math. Soc., Providence, RI, 2013, 3-33.
  10. I. Cho, P.E.T. Jorgensen, Semicircular elements induced by p-adic number fields, Opuscula Math. 37 (2017) 5, 665-703.
  11. H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993.
  12. H. Davenport, Multiplicative Number Theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000.
  13. P. Deligne, Applications de la formule des traces aux sommes trigonometriques, Cohomologies étale, Lecture Notes in Math., vol. 569, Springer, Berlin, 1977, 168-232.
  14. G. Feldman, M. Myronyuk, On a characterization of idempotent distributions on discrete fields and on the field of \(p\)-adic numbers, J. Theoret. Probab. 30 (2017) 2, 608-623.
  15. T. Gillespie, Superposition of zeroes of automorphic \(L\)-functions and functoriality, PhD Thesis, University of Iowa, (2010).
  16. T. Gillespie, GuangHua Ji, Prime number theorems for Rankin-Selberg \(L\)-functions over number fields, Sci. China Math. 54 (2011) 1, 35-46.
  17. K. Girstmair, Dedekind sums in the p-adic number field, Int. J. Number Theory 14 (2018) 2, 527-533.
  18. P. Ingram, \(p\)-adic uniformization and the action of Galois on certain affine correspondences, Canad. Math. Bull. 61 (2018) 3, 531-542.
  19. H. Iwaniec, E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004.
  20. C. Jantzen, Duality for classical \(p\)-adic groups: the half-integral case, Representation Theory 22 (2018), 160-201.
  21. T. Kemp, R. Speicher, Strong Haagerup inequalities for free \(\mathcal{R}\)-diagonal elements, J. Funct. Anal. 251 (2007) 1, 141-173.
  22. F. Larsen, Powers of R-diagonal elements, J. Operator Theory 47 (2002) 1, 197-212.
  23. N. Mariaule, The field of \(p\)-adic numbers with a predicate for the powers of an integer, J. Symb. Log. 82 (2017) 1, 166-182.
  24. A. Nica, D. Shlyakhtenko, R. Speicher, \(R\)-diagonal elements and freeness with amalgamation, Canad. J. Math. 53 (2001) 2, 355-381.
  25. F. Radulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994) 2, 347-389.
  26. M. Solleveld, Topological \(K\)-theory of affine Hecke algebras, Ann. K-Theory 3 (2018) 3, 395-460.
  27. R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627.
  28. V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics, Series on Soviet and East European Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
  29. D. Voiculescu, K. Dykema, A. Nica, Free Random Variables. A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992.
  • 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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