Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

Trans. Amer. Math. Soc.

2019

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Abstract. Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq 2 /n goes to zero, and not so if (p, q) sits on the curve pq 2 = σn. A previous work by Jiang [17] shows that the total variation distance goes to zero if both p/ √ n and q/ √ n go to zero, and it is not true provided p = c √ n and q = d √ n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ.

Section: Introductionmentioning

confidence: 84%

“…As mentioned earlier, the work [17] has been applied to other random matrix problems [18], the wireless communications [24,25,26] and a problem from Big Data [5]. In this paper we consider other three probability metrics: the Hellinger distance, the Kullback-Leibler distance and the Euclidean distance.…”

Section: Remarks and Future Questionsmentioning

confidence: 99%

Plot of CLT

Trans. Amer. Math. Soc.

2019

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“…In literature, a Jacobi matrix is also called a MANOVA matrix, which has been used extensively in the multivariate analysis from the field of Statistics. Some applications of the truncations of Haar-invariant matrices can be found in, for example, Eaton (1989), Diaconis, Eaton and Lauritzen (1992) and Jiang (2009).…”

Section: Introductionmentioning

confidence: 99%

Circular law and arc law for truncation of random unitary matrix

Dong

Journal of Mathematical Physics

2012

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Let V be the m × m upper-left corner of an n × n Haar-invariant unitary matrix. Let λ 1 , · · · , λ m be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ 1 , · · · , λ m goes to the circular law, that is, the uniform distribution on {z ∈ C; |z| ≤ 1} as m → ∞ with m/n → 0. We also prove that the empirical distribution of λ 1 , · · · , λ m goes to the arc law, that is, the uniform distribution on {z ∈ C; |z| = 1} as m/n → 1. These explain two observations byŻyczkowski and Sommers (2000).

“…This connection is established recently by Collins [6]. For other results in this realm, one can see [1,7,12,14,17,30].…”

Section: Introductionmentioning

confidence: 65%

“…normal random vectors. This method is used to study the entries of matrices which generate the Haar measures on O(n) and U (n), and then is applied for investigating the properties of eigenvalues of Jacobi matrices, see [15,16,17]. There are other ways to generate Haar measures on the classical compact groups, see e.g., Mezzadri [29].…”

Section: Introductionmentioning

confidence: 99%

A variance formula related to a quantum conductance problem

2009

Physics Letters A

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Let t be a block of an Haar-invariant orthogonal (β = 1), unitary (β = 2) or symplectic (β = 4) matrix from the classical compact groups O(n), U (n) or Sp(n), respectively. We obtain a close form for V ar(tr(t * t)). The case for β = 2 is related to a quantum conductance problem, and our formula recovers a result obtained by several authors. Moreover, our result shows that the variance has a limit (8β) −1 for β = 1, 2 and 4 as the sizes of t go to infinity in a special way.Although t in our formulation comes from a block of an Haar-invariant matrix from the classical compact groups, the above limit is consistent with a formula by Beenakker, where t is a block of a circular ensemble.