Genus Expansion for Real Wishart Matrices

Abstract

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.

Real Second-Order Freeness and the Asymptotic Real Second-Order Freeness of Several Real Matrix Models

“…A premap is the appropriate object to describe the faces, hyperedges, or vertices of an unoriented surface: the conditions ensure the consistency of the description on an orientable two-sheeted covering space (see, for example, [13], pages 234-235, for the construction of this two-sheeted cover). See [7,14,16,20,22] for more information on how surfaced hypergraphs can be represented as sets of permutations. This interpretation is often useful for understanding subsequent calculations, but is not necessary to the proofs.…”

“…Vertex information on the covering space is given by the permutation γ + π −1 γ −1 − , and the traces of the Y k matrices are given by its inverse. See [14,16,20,25] for examples of how matrix integrals may be interpreted in terms of surfaced hypergraphs. The definition of the Euler characteristic of γ and π which follows is the natural one in this interpretation ( [7] gives a compatible definition of genus).…”

“…This is possible because the transposes appear in a regular pattern; in some sense, sign takes the place of parity in this expression. Geometrically, this calculation can be represented as in [20], except we consider the alternate vertices which here contain the Y k matrices as hyperedges, encoded in the permutation π, and the vertices induced in this unoriented hypermap are the vertices containing the Y k matrices. An example with a diagram is given after the proof.…”

“…If we wish to calculate the value of Each term in the sum can be represented as a face gluing (see [20], Section 3, for more detail). The pairing ρ = (1, 5) (2, 7) (3, 9) (4, 10) (6,8) is shown in Figure 2, left.…”

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“…Other interesting mathematical results for the Laguerre case can be found e.g. in [23] and references therein. We are also aware that formulae for the Wishart-Laguerre and Jacobi moments for β = 1, 2, 4 and finite N have been derived by Mezzadri and Simm [24].…”