Poisson operator on the interacting Fock space associated with a discrete-time quantum walk

Abstract

We study the Poisson operator on the interacting Fock space associated with a discrete-time quantum walk, which we call the QW-Poisson operator. First, we investigate the spectral properties of the QW-Poisson distribution. In particular, we establish a relation between the spectral distributions of the Poisson operator and the reversed Poisson operator on a general interacting Fock space via a size-biased transform. Next, we study the edge behavior of the density of the QW-Poisson distribution. We show that a phase transition occurs at the left endpoint of the support: depending on the parameter, the density either decays to $0$ or blows up to $+\infty$. Moreover, this phase transition coincides with the transition in the number of atoms of the QW-Poisson distribution, equivalently, in the point spectrum of the QW-Poisson operator, and with whether $0$ belongs to the spectrum of the QW-Poisson operator. We then compute the moment-generating function and several statistical quantities of the QW-Poisson operator. We also obtain a limit theorem for the Konno distribution through a Poisson approximation. Finally, we study a connection between the interacting Fock space associated with a discrete-time quantum walk and noncommutative probability theory.

40 pages. Revised the discussion of previous work and new perspectives on the connections between quantum walks and noncommutative probability theory; revised the statement of Theorem 4.1