Quantum Decomposition and Quantum Central Limit Theorem

On the basis of the canonical relation between an interacting Fock space and a system of orthogonal polynomials we introduce the notion of quantum decomposition of a real random variable in an algebraic probability space. To understand prototypes we review some basic examples appearing from the Boson, Fermion, and free Fock spaces. We then prove quantum central limit theorems for the quantum components of the adjacency matrices (combinatorial Laplacians) of a growing family of regular connected graphs. As a corollary, asymptotic properties of the adjacency matrix are obtained. Concrete examples include lattices, homogeneous trees, Cayley graphs of the Coxeter groups, Hamming graphs and Johnson graphs. In particular, asymptotic spectral distribution of the adjacency matrix of a Johnson graph is described by an interacting Fock space corresponding to the Meixner polynomials which are one-parameter deformation of the Laguerre polynomials.