Noncrossing Partitions, Catalan Words, and the Semicircle Law

Abstract

As is well known, the joint limit distribution of independent Wigner matrices is free with the marginals being semicircular. This freeness is intimately tied to noncrossing pair partitions or, equivalently what are known as Catalan words, each of which contributes one to the limit moments. We investigate the following questions. Consider a sequence of patterned matrices: (i) When do only Catalan words contribute (one), so that we get the semicircle limit? (ii) When does each Catalan word contribute one (with possible nonzero contribution from non-Catalan words)? (iii) For what matrix models do Catalan words not necessarily contribute one each and non-semicircle limits arise, even when non-Catalan words have zero contribution? In particular we show that in a general sense, the semicircle law serves as a lower bound for possible limits. Further, there is a large class of non-Wigner matrices whose limit is the semicircle. This may be viewed as robustness of the semicircle law. Similarly, there is a large class of block matrices whose limit is not semicircular.