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Asymptotic joint spectra of Cartesian powers of strongly regular graphs and bivariate Charlier–Hermite polynomials

Tom 162 / 2020

John Vincent S. Morales, Nobuaki Obata, Hajime Tanaka Colloquium Mathematicum 162 (2020), 1-22 MSC: Primary 46L53; Secondary 60F05, 05E30, 33C45. DOI: 10.4064/cm7724-7-2019 Opublikowany online: 1 April 2020

Streszczenie

Generalizing previous work of Hora (1998) on the asymptotic spectral analysis for the Hamming graph $H(n,q)$ which is the $n$th Cartesian power $K_q^{\square n}$ of the complete graph $K_q$ on $q$ vertices, we describe the possible limits of the joint spectral distribution of the pair $(G^{\square n},\overline {G}{}^{\square n})$ of the $n$th Cartesian powers of a strongly regular graph $G$ and its complement $\overline {G}$, where we let $n\rightarrow \infty $, and $G$ may vary with $n$. This result is an analogue of the bivariate central limit theorem, and we obtain in this way the bivariate Poisson distributions and the standard bivariate Gaussian distribution, together with the product measures of univariate Poisson and Gaussian distributions. We also report a family of bivariate hypergeometric orthogonal polynomials with respect to the last distributions, which we call the bivariate Charlier–Hermite polynomials, and prove basic formulas for them. This family of orthogonal polynomials seems previously unnoticed, possibly because of its peculiarity.

Autorzy

  • John Vincent S. MoralesMathematics and Statistics Department
    College of Science
    De La Salle University
    Manila 1004, Philippines
    e-mail
  • Nobuaki ObataResearch Center for Pure and
    Applied Mathematics
    Graduate School of Information Sciences
    Tohoku University
    Sendai 980-8579, Japan
    e-mail
  • Hajime TanakaResearch Center for Pure and Applied Mathematics
    Graduate School of Information Sciences
    Tohoku University
    Sendai 980-8579, Japan
    e-mail

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