Random unitaries in non-commutative tori, and an asymptotic model for q-circular systems

Abstract

We consider the concept of q-circular system, which is a deformation of the circular system from free probability, taking place in the framework of the so-called “qcommutation relations”. We show that certain averages of random unitaries in noncommutative tori behave asymptotically like a q-circular system. More precisely: let q be in (−1, 1); let s, k be positive integers; let (ρij)1≤i<j≤ks be independent random variables with values in the unit circle, such that ∫ ρij = q, ∀ 1 ≤ i < j ≤ ks; and let U1, . . . , Uks be random unitaries such that UiUj = ρijUjUi, ∀1 ≤ i < j ≤ ks. If we set: Xr := 1 √ k ( Ur + Ur+s + · · · + Ur+(k−1)s ), 1 ≤ r ≤ s, then the family X1, . . . , Xs behaves for k → ∞ like a q-circular system with s elements. The above result generalizes to the case when instead of the hypothesis “ ∫ ρij = q” we start with “ ∫ ρij = z”, where z is a complex number such that |z| < 1. In this case the limit distribution of X1, . . . , Xs is what we call a z-circular system. From the combinatorial point of view, the new feature brought in by a z-circular system is that its description involves the enumeration of oriented crossings of certain pairings; it is only in the case when z = z = q that the orientations cancel out, allowing the q-circular system to be described via non-oriented crossings. As a consequence of the result, one can easily construct families of random matrices which converge in distribution to q-circular (or more generally z-circular) systems. ∗Research supported by a grant from the Natural Sciences and Engineering Research Council, Canada.