On the Diaconis-Shahshahani Method in Random Matrix Theory

Stolz, Michael

doi:10.1007/s10801-005-4629-x

Cited by 25 publications

(12 citation statements)

References 8 publications

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“…the representation ρ k , and s ∈ S k has cycle type (1 a 1 2 a 2 · · · r ar ), whence k = r j=1 ja j . See [16] for details. Now we turn to the problem whose solution in [4] is the basic tool for our present purposes.…”

Section: Notation and Statement Of The Main Resultsmentioning

confidence: 99%

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Borel theorems for random matrices from the classical compact symmetric spaces

Collins¹,

Stolz²

2008

Ann. Probab.

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We study random vectors of the form (Tr(A (1) V ), . . . , Tr(A (r) V )), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A (ν) are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins andŚniady, for polynomial functions on the classical compact groups.

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Section: Notation and Statement Of The Main Resultsmentioning

confidence: 99%

“…For n ≥ l, it follows from Weyl's First Fundamental Theorem and Section 3.4 of[16] that{θ m o : m o ∈ M(2l)} is a basis for [V ⊗2l ] Sp 2n , where M(2l) is identified with the set of all ordered pair partitions {(m ν , n ν ) : ν = 1, . .…”

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confidence: 99%

Borel theorems for random matrices from the classical compact symmetric spaces

Collins¹,

Stolz²

2008

Ann. Probab.

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“…The value distributions of traces of random unitary matrices have been studied extensively over the past fifteen years [9,13,15,24,11,14,21,25]. The main reason is that they are connected with the linear statistics S N (χ) := χ e iθ 1 + · · · + χ e iθ N , (1.1) where χ is a suitable test function and e iθ 1 , .…”

Section: Introductionmentioning

confidence: 99%

Rate of convergence of linear functions on the unitary group

Keating¹,

Mezzadri²,

Singphu³

2010

J. Phys. A: Math. Theor.

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We study the rate of convergence to a normal random variable of the real and imaginary parts of Tr A N U , where U is an N × N random unitary matrix and A N is a deterministic complex matrix. We show that the rate of convergence is O N −2+b , with 0 ≤ b < 1, depending only on the asymptotic behaviour of the singular values of A N ; for example, if the singular values are non-degenerate, different from zero and O(1) as N → ∞, then b = 0. The proof uses a Berry-Esséen inequality for linear combinations of eigenvalues of random unitary matrices, and so appropriate for strongly dependent random variables. MSC: 15B52, 60F05

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“…Both Stolz [23] and Rains [18] have already used the same group for this computation. On page 1287, we highlight the crucial features that B k satisfies and make the proofs work.…”

Section: Introductionmentioning

confidence: 99%

“…We would like to point out that our proof of Theorem 1 involves the hyperoctahedral group B k . Both Stolz [22] and Rains [17] have already used the same group for this computation.…”

Section: Introductionmentioning

confidence: 99%

Averages over classical Lie groups, twisted by characters

Dehaye

2007

Journal of Combinatorial Theory, Series A

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We compute E G ( i tr(g λ i )), where g ∈ G = Sp(2n) or SO(m) (m = 2n, 2n+1) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49-62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E G Φ n are affected when we introduce a character χ G λ into the integrand. We show that the value of E G χ G λ Φ n /E G Φ n approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Φ n and λ and is independent of the Cartan type of G. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252-271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E G Φ n due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519-545] and provide asymptotics for E G χ G λ Φ n .

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On the Diaconis-Shahshahani Method in Random Matrix Theory

Cited by 25 publications

References 8 publications

Borel theorems for random matrices from the classical compact symmetric spaces

Borel theorems for random matrices from the classical compact symmetric spaces

Rate of convergence of linear functions on the unitary group

Averages over classical Lie groups, twisted by characters

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