arXiv subjects

C. Emily I. Redelmeier

Publications and source records attributed to C. Emily I. Redelmeier.

4 recordsLinked to original sources

arXiv2012

Appendix to "Real second-order freeness and the asymptotic real second-order freeness of several real matrix models": Examples and Diagrams

We present examples and diagrams illustrating the proofs appearing in "Real second-order freeness and the asymptotic real second-order freeness of several real matrix models", to which this paper is meant to be an appendix. We show how matrix calculations may be represented by topological surface gluings, which may in turn be represented as permutations.

math.PR↗

arXiv2012

Real second-order freeness and the asymptotic real second-order freeness of several real matrix ensembles

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, three real models of random matrices, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. We use a combinatorial approach to the matrix calculations similar to the genus expansion for complex random matrices, but in which nonorientable surfaces appear, demonstrating the commonality between the real models and the distinction from their complex analogues, motivating this distinct definition. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears.

math.OA↗

arXiv2010

Representations of the Temperley-Lieb Algebra via a New Inner Product on Half-Diagrams

We describe an inner product on the diagrams on which the Temperley-Lieb algebra can be represented. We exhibit several constructions which are in natural combinatorial bijection with these diagrams, which are generalizations of various constructions counted by the Catalan numbers. We use a method similar to the existing ones for orthogonalizing the Temperley-Lieb algebra to construct an orthogonal basis for the vector space over these diagrams.

math.CO↗

arXiv2009

Genus expansion for real Wishart matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central limit-type theorem for the asymptotic distribution around the expected value.

math.PR↗