One-mode interacting Fock spaces and random walks on graphs

Abstract

A one-mode interacting Fock space is reformulated in a slightly generalized form in terms of a tridiagonal matrix. We derive the continued fraction expansion of its resolvent and the combinatorial part of the Accardi–Bożejko formula. Under certain positivity condition, a probability distribution on the real line is related and the Karlin–McGregor formula is reproduced under slightly weaker conditions. We show concrete computation for random walks on graphs, e.g. the free Meixner law is derived from a random walk on a spidernet.

Keywords:

• interacting Fock space
• orthogonal polynomials
• Karlin–McGregor formula
• Accardi–Bożejko formula
• random walk

2000 Mathematics Subject Classification::

• primary: 81S25
• secondary: 05C81, 42C05