arXiv subjects

C. E. I. Redelmeier

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

5 recordsLinked to original sources

arXiv2020

Real and quaternionic second-order free cumulants and connections to matrix cumulants

We present definitions for real and quaternionic second-order free cumulants, functions whose collective vanshing when applied to elements from different subalgebras is equivalent to the second-order real (resp.\ quaternionic) freeness of those subalgebras. We construct a poset related to the annular noncrossing partitions, and calculate the Möbius function, which may be used to compute the coefficients in the expression for the second-order free cumulants. We show the connection between second-order free cumulants and the topological expansion interpretation of matrix cumulants. This provides a construction for higher-order free cumulants. Coefficients are given in terms of the asymptotics of the cumulants of the Weingarten function.

math.PR↗

arXiv2020

Möbius Functions of Some Annular Noncrossing Objects

We present the Möbius functions of several posets of annular noncrossing objects, namely a self-dual extension of the annular noncrossing permutations, minimal length annular partitioned permutations, and annular noncrossing partitons.

math.CO↗

arXiv2015

Topological expansion for Haar-distributed orthogonal matrices and second-order freeness of orthogonally invariant ensembles

We present a genus expansion-type expression for the expected values of products of traces of expressions involving Haar-distributed orthogonal matrices. As with other real genus expansions, nonorientable surfaces appear, in addition to the orientable surfaces of the complex expansion. We use this expression to demonstrate that independent random matrices which are orthogonally in general position, such as matrices whose distributions are orthogonally invariant, are asymptotically real second-order free.

math.PR↗

arXiv2015

Quaternionic Second-Order Freeness and the Fluctuations of Large Symplectically Invariant Random Matrices

We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behaviour of unitarily invariant and orthogonally invariant random matrices respectively. Most notably, because the quaternionic trace does not have the cyclic property of a trace over a commutative field, the asymmetries which appear in the multi-matrix context result in an asymmetric contribution from the terms which appear symmetrically in the complex and real cases.

math.FA↗

arXiv2015

Explicit Multi-Matrix Topological Expansion for Quaternionic Random Matrices

We present an explicit formula for the expected value of a product of several independent symplectically invariant matrices in which the trace and real part function may be applied, possibly to different subexpressions. This takes the form of a topological expansion; however, each term has two topologies: one for the trace, and another for the real part. The traces and real parts can always be written in terms of index contraction, but in some cases, it is possible to write the expression as a product in which the two functions are applied to bracketed intervals in a legal bracket diagram. We present the conditions under which this may be done, and an algorithm to construct such an expression given the contracted indices when possible. The summands in the topological expansion are written in terms of matrix cumulants. We compute the matrix cumulants of quaternionic Ginibre, Gaussian symplectic, quaternionic Wishart, and Haar-distributed symplectic matrices, which allow direct computation of an expression constructed from several independent ensembles of any of these matrices.

math.PR↗