68,046 research outputs found

    Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices

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    We consider random matrices of the form H=W+λVH = W + \lambda V, λ∈R+\lambda\in\mathbb{R}^+, where WW is a real symmetric or complex Hermitian Wigner matrix of size NN and VV is a real bounded diagonal random matrix of size NN with i.i.d.\ entries that are independent of WW. We assume subexponential decay for the matrix entries of WW and we choose λ∼1\lambda \sim 1, so that the eigenvalues of WW and λV\lambda V are typically of the same order. Further, we assume that the density of the entries of VV is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈R+\lambda_+\in\mathbb{R}^+ such that the largest eigenvalues of HH are in the limit of large NN determined by the order statistics of VV for λ>λ+\lambda>\lambda_+. In particular, the largest eigenvalue of HH has a Weibull distribution in the limit N→∞N\to\infty if λ>λ+\lambda>\lambda_+. Moreover, for NN sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+\lambda>\lambda_+, while they are completely delocalized for λ<λ+\lambda<\lambda_+. Similar results hold for the lowest eigenvalues.Comment: 47 page

    Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,−2m)J(2n,-2m) knot orbifolds

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    We calculate the Chern-Simons invariants of the hyperbolic J(2n,−2m)J(2n,-2m) knot orbifolds using the Schl\"{a}fli formula for the generalized Chern-Simons function on the family of cone-manifold structures of J(2n,−2m)J(2n,-2m) knot. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee's methods to a bi-infinite family. We dealt with even slopes just as easily as odd ones. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic J(2n,−2m)J(2n,-2m) knot orbifolds. For the fundamental group of J(2n,−2m)J(2n, -2m) knot, we take and tailor Hoste and Shanahan's. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert's canonical 2-bridge diagram or not.Comment: 9 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1601.00723, arXiv:1607.0804
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