Poisson type operators on the Fock space of type B and in the Blitvi\'c model
Abstract
In Bia97 Biane proposed a new statistic on set partitions which he called restricted crossings. In a series of papers Ans01,Ans04,Ans04b,Ans05 Anshelevich showed that this statistic is an essential tool to investigate stochastic processes on q-Fock space. In particular, Anshelevich constructed operators whose moments count restricted crossings and used these operators to develop a beautiful theory of noncommutative q-L\'evy processes. In the present paper following Anshelevich we define gauge operators on (α,q)-Fock and cumulants which are governed by statistics on partitions of type B. In addition we investigate this construction in the context of a model of Blitvi\'c model B12, where some related but different combinatorial structures appear, and we explain their relation with t-free probability.
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