Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law

We study the distribution (with respect to the vacuum state) of a family of partial sums S 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.