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Two-parameter Non-commutative Central Limit Theorem

The non-commutative Central Limit Theorem (CLT) introduced by Speicher in 1992 states that given almost any sequence of non-commutative random variables that commute or anti-commute pair-wise, the *-moments of the normalized partial sum S_N=(b_1+...+ b_N)/\sqrt{N} are given by a Wick-type formula refined to count the number of crossings in the underlying pair-partitions. When coupled with explicit matrix models, the theorem yields random matrix models for creation and annihilation operators on the q-Fock space of Bozejko and Speicher. In this paper, we derive a non-commutative CLT when the pair-wise commutation coefficients are real numbers (as opposed to signs). The statistics of the limiting random variable are a second-parameter refinement of those above, jointly indexing the number of crossings and nestings in the underlying pair-partitions. Coupled with analogous matrix constructions, the theorem yields random matrix models for creation and annihilation operators on the recently introduced (q,t)-Fock space.