Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law

TL;DR

This paper explores the distribution of sums of position operators in weakly monotone Fock space, revealing their relation to monotone independence and providing explicit and recursive descriptions of their distributions.

Contribution

It establishes that these sums are monotone independent and connects their distribution to the m-fold monotone convolution of the semicircle law, with explicit results for small m.

Findings

Single operators follow the Wigner law.

Families of operators are monotone independent.

Explicit distribution for m=2, recursive moments for m>2.

Abstract

We study the distribution (w.r.t. the vacuum state) of family of partial sums Sm of position operators on weakly monotone Fock space. We show that any single operator has the Wigner law, and an arbitrary family of them (with the index set linearly ordered) is a collection of monotone independent random variables. It turns out that our problem equivalently consists in finding the m-fold monotone convolution of the semicircle law. For m = 2 we compute the explicit distribution. For any m > 2 we give the moments of the measure, and show it is absolutely continuous and compactly supported on a symmetric interval whose endpoints can be found by a recurrence relation.