- journal article metadata
European Mathematical Society Publishing House
2016-12-25 23:45:00
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2016
4
The arcsine law and an asymptotic behavior of orthogonal polynomials
Hayato
Saigo
Nagahama Institute of Bio-Science and Technology, NAGAHAMA, JAPAN
Hiroki
Sako
Niigata University, NIIGATA, JAPAN
Noncommutative probability arcsine law, interacting Fock space
Interacting Fock spaces connect the study of quantum probability theory, classical random variables, and orthogonal polynomials. They are pre-Hilbert spaces associated with creation, preservation, and annihilation processes. We prove that if three processes are asymptotically commutative, the arcsine law arises as the “large quantum number limits.” As a corollary, it is shown that for many probability measures, the asymptotic behavior of orthogonal polynomials is described by the arcsine function. A weaker form of asymptotic commutativity provides us with a discretized arcsine law, which is described by the Bessel functions of the fi rst kind.
Functional analysis
Special functions
Probability theory and stochastic processes
Quantum theory
405
427
10.4171/AIHPD/34
http://www.ems-ph.org/doi/10.4171/AIHPD/34