Distribution for Non Symmetric Position Operators on the Free Toy Fock Space and Its Approximation on the Full Fock Space

Abstract

In this paper, we investigate the “Poisson-type” limit theorem with respect to the vacuum state of a family of partial sums of non-symmetric position operators under an appropriate scaling on the free toy Fock space. We give a formula for the vacuum moment in relation with the combinatorics of non-crossing partitions. We show that the asymptotic measure associated to the limit of the partial sums of these operators is the free Meixner law with an atomic and an absolutely continuous part, whereas the probability distribution of any single operator is the two-point probability. The approximation of such operators on the full Fock space is given.