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A recursion formula for the moments of the Gaussian orthogonal ensemble. (English) Zbl 1184.60003

There exists a three-term-recurrence relation for the moments of \(Tr(X^N)\) for the \(N\)-dimensional Gaussian Unitary Ensembles (GUE) due to J. Harer and D. Zagier [Invent. Math. 85, 457–485 (1986; Zbl 0616.14017)]. This recurrence can be proved either by combinatorial arguments or by Hermite polynomial calculus.
In the paper under review, the author derives an analogue for Gaussian Orthogonal Ensembles (GOE) in the form of a five-term-recurrence. The proof is based on Gaussian integration by parts and differential equations for Laplace transforms. Similar as for the result of Harer and Zagier, this result admits a combinatorial interpretation via numbers of vertex maps in locally orientable surfaces with a given number of edges and faces. Moreover, the author presents a corresponding recurrence for Gaussian Symplectic Ensembes (GSE) by using a duality argument between GOE and GSE due to Mulase and Waldron. These five-term-recurrence formulas are used in the end to obtain sharp bounds for tails of the largest eigenvalue of GOE and, by moment comparison, of more general families of Wigner matrices.

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
60E05 Probability distributions: general theory

Citations:

Zbl 0616.14017
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Online Encyclopedia of Integer Sequences:

a(n) = number of maximal chord diagrams by genus d^(||)_2n.

References:

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