Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law

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.