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Universality properties of Gelfand-Tsetlin patterns. (English) Zbl 1271.60017

For every \(n\in\mathbb N\), let \(A_n\) be a random Hermitian \(n\times n\)-matrix whose distribution is unitarily invariant. For each \(r=1,\dotsc,n\), let \(\pi_r\) be the diagonal projection matrix of rank \(r\) with the diagonal \((1,\dotsc,1,0,\dotsc,0)\). The author studies the asymptotic behavior of the eigenvalues of \(B_n:=\pi_{q_n} A_n \pi_{q_n}\) (which is a \(q_n\times q_n\) submatrix of \(A_n\)) under the condition that \(q_n/n\) tends to a limit \(\alpha\in (0,1)\), as \(n\to\infty\). More precisely, under appropriate assumptions, the author proves that the correlation kernel associated to the eigenvalues of \(B_n\) converges to the sine kernel in the bulk of the limiting spectral measure. One special case is that of the Gaussian unitary ensemble (GUE). In this case, the joint distribution of the eigenvalues of all minors of \(A_n\) is known to be a determinantal point process on \(\{1,\dotsc,n\}\times \mathbb R\) with certain explicit correlation kernel (see [K.Johansson and E.Nordenstam, Electron. J. Probab. 11, 1342–1371 (2006; Zbl 1127.60047)]). Another special case can be obtained by taking \(A_n=U_n C_n U_n\), where \(C_n\) is a fixed Hermitean matrix and \(U_n\) is a random unitary matrix chosen according to the Haar measure. The eigenvalues of \(B_n\) can be then interpreted as the \(q_n\)-th level of a Gelfand-Tsetlin pattern chosen uniformly from the set of all Gelfand-Tsetlin patterns whose \(n\)-th level is fixed.

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)

Citations:

Zbl 1127.60047
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