Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators

TL;DR

This paper analyzes the vacuum distribution of nonsymmetric position operator sums on weakly monotone and monotone Fock spaces, revealing their connection to free Meixner laws and Poisson-type limit laws through combinatorial and asymptotic analysis.

Contribution

It introduces a recursive formula for vacuum moments of nonsymmetric position operator sums and characterizes their distributions as monotone convolutions of free Meixner laws.

Findings

Operators have vacuum laws in the free Meixner class.

Asymptotic sums exhibit Poisson-type limit laws within the free Meixner class.

Distribution includes atomic and absolutely continuous parts.

Abstract

We study the vacuum distribution, under an appropriate scaling, of a family of partial sums of nonsymmetric position operators on weakly monotone and monotone Fock spaces, respectively. We preliminary treat the case of weakly monotone Fock space, and show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any finite sum. Since the operators are monotone independent, the distribution is the monotone convolution of the free Meixner law above. We also investigate the asymptotic measure for these sums, which can be seen as "Poisson type" limit law. It turns out to belong to the free Meixner class, with an atomic and an absolutely continuous part (w.r.t. the Lebesgue measure). Finally, we briefly apply analogous considerations to the case of monotone Fock space.