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Title:  Real SecondOrder Freeness and Fluctuations of Random Matrices 
Authors:  REDELMEIER, CATHERINE EMILY ISKA 

Keywords:  random matrices central limit theorem secondorder freeness free probability 
Issue Date:  9Sep2011 
Series/Report no.:  Canadian theses 
Abstract:  We introduce real secondorder freeness in secondorder noncommutative probability spaces. We demonstrate that under this definition, independent ensembles of the three real models of random matrices which we consider, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically secondorder free. These ensembles do not satisfy the complex definition of secondorder freeness satisfied by their complex analogues. This definition may be used to calculate the asymptotic fluctuations of products of matrices in terms of the fluctuations of each ensemble.
We use a combinatorial approach to the matrix calculations similar to genus expansion, but in which nonorientable surfaces appear, demonstrating the commonality between the real ensembles and the distinction from their complex analogues, motivating this distinct definition. We generalize the description of graphs on surfaces in terms of the symmetric group to the nonorientable case.
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. 
Description:  Thesis (Ph.D, Mathematics & Statistics)  Queen's University, 20110909 11:07:37.414 
URI:  http://hdl.handle.net/1974/6711 
Appears in Collections:  Mathematics & Statistics Graduate Theses Queen's Theses & Dissertations

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