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.
Highlights
The present notes are a continuation of the investigation started by the authors in [8]
There we studied the vacuum distribution of sums of symmetric position operators on the weakly monotone Fock space (WM-Fock for short), based on a separable Hilbert space H
The operators above are called gaussian since they realize, in suitable Hilbert spaces, natural examples of noncommutative random variables whose distributions, w.r.t. a distinguished vector state, are central limit laws. It was observed by Hudson and Parthasarathy [15] that the classical Brownian motion can be realized on the symmetric Fock space as the sum of creation and annihilation processes
Summary
The present notes are a continuation of the investigation started by the authors in [8]. 2 we present the WM-Fock space with creation, annihilation and preservation operators, and recall basic information about labeled noncrossing partitions, combinatorics of Motzkin and Riordan paths, basic properties of the Cauchy transform of a probability measure, and free Meixner laws as well. As. nonsymmetric position operators with intensity λ are monotone independent random variables in the ∗-algebra generated by weakly monotone creation operators (see [8]), one naturally tries to get some information on the vacuum law of their partial sums mk=1(Gk + λ A0k), which correspond to the m-fold monotone convolution of the free. The vacuum law of a single nonsymmetric position operator here is a two-points discrete measure, the Poisson type limit is the same as in the weakly monotone case
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