Some combinatorial aspects of the spectra of normally distributed random matrices

Hanlon, Philip J.; Stanley, Richard P.; Stembridge, John R.

doi:10.1090/conm/138/1199126

Cited by 88 publications

(127 citation statements)

References 6 publications

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“…In the constructions used in [3,9,10], two graphs are constructed whose union consists of cycles whose lengths are 2n 1 , . .…”

Section: Geometric Motivationmentioning

confidence: 99%

“…This matrix ensemble was introduced in Wishart's paper of 1928 [18] in order to study the expected covariances of multivariate data. The moments of Wishart matrices, as well as more general symmetric polynomials in their eigenvalues, are studied in such papers as [3,5,6,9,10]. The behaviour of these quantities in large matrices and the connections to free probability are considered in papers such as [5,6,15].…”

Section: Introductionmentioning

confidence: 99%

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Genus Expansion for Real Wishart Matrices

Redelmeier

2010

J Theor Probab

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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.

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“…In the constructions used in [3,9,10], two graphs are constructed whose union consists of cycles whose lengths are 2n 1 , . .…”

Section: Geometric Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Genus Expansion for Real Wishart Matrices

Redelmeier

2010

J Theor Probab

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“…Lemma 3.4 while known, is perhaps not well known, and since analogous results will be needed in later sections, a proof along the lines of one in [19] is included. …”

Section: Lemma 32 Let ν and χ Be Irreducible Characters Of A Finitementioning

confidence: 99%

“…constant on double cosets of K in G), let p m (K r ) denote the probability that the m-fold convolution of P assigns to the double coset Kg r K. Lemma 4.4 is an analog of Lemma 3.4 and could be proved along similar lines, as in [19]. Instead, we use the language of Fourier analysis, as developed on page 395 of [29].…”

Section: Lemma 42 ([29] Page 389) Formentioning

confidence: 99%

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Stein's method and random character ratios

Fulman

2008

Trans. Amer. Math. Soc.

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Abstract. Stein's method is used to prove limit theorems for random character ratios. Tools are developed for four types of structures: finite groups, Gelfand pairs, twisted Gelfand pairs, and association schemes. As one example an error term is obtained for a central limit theorem of Kerov on the spectrum of the Cayley graph of the symmetric group generated by i-cycles. Other main examples include an error term for a central limit theorem of Ivanov on character ratios of random projective representations of the symmetric group, and a new central limit theorem for the spectrum of certain random walks on perfect matchings. The results are obtained with very little information: a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk. The limit theorems stated in this paper are for normal approximation, but many of the tools developed are applicable for arbitrary distributional approximation.

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MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions

Richards

2001

Advances in Applied Mathematics

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) independently derived a generalization of MacMahon's master theorem. In this article we apply their result to obtain an explicit formula for the moments of arbitrary polynomials in the entries of X, a real random matrix having a Wishart distribution. In the case of the complex Wishart distributions, the same method is applicable. Furthermore, we apply the representation theory of GL d , the complex general linear group, to derive explicit formulas for the expectation of Kronecker products of any complex Wishart random matrix.  2001 Elsevier Science

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Some combinatorial aspects of the spectra of normally distributed random matrices

Abstract: This paper is in h a l form and no version of it will be submitted for publication elsewhere.

Cited by 88 publications

References 6 publications

Genus Expansion for Real Wishart Matrices

Genus Expansion for Real Wishart Matrices

Stein's method and random character ratios

MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions

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