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Please use this identifier to cite or link to this item: http://hdl.handle.net/1974/6711

Title: Real Second-Order Freeness and Fluctuations of Random Matrices

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Keywords: random matrices
central limit theorem
second-order freeness
free probability
Issue Date: 9-Sep-2011
Series/Report no.: Canadian theses
Abstract: We introduce real second-order freeness in second-order 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 second-order free. These ensembles do not satisfy the complex definition of second-order 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, 2011-09-09 11:07:37.414
URI: http://hdl.handle.net/1974/6711
Appears in Collections:Queen's Graduate Theses and Dissertations
Department of Mathematics and Statistics Graduate Theses

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