68,046 research outputs found
Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices
We consider random matrices of the form ,
, where is a real symmetric or complex Hermitian
Wigner matrix of size and is a real bounded diagonal random matrix of
size with i.i.d.\ entries that are independent of . We assume
subexponential decay for the matrix entries of and we choose , so that the eigenvalues of and are typically of the same
order. Further, we assume that the density of the entries of is supported
on a single interval and is convex near the edges of its support. In this paper
we prove that there is such that the largest
eigenvalues of are in the limit of large determined by the order
statistics of for . In particular, the largest
eigenvalue of has a Weibull distribution in the limit if
. Moreover, for sufficiently large, we show that the
eigenvectors associated to the largest eigenvalues are partially localized for
, while they are completely delocalized for
. Similar results hold for the lowest eigenvalues.Comment: 47 page
Explicit formulae for Chern-Simons invariants of the hyperbolic knot orbifolds
We calculate the Chern-Simons invariants of the hyperbolic knot
orbifolds using the Schl\"{a}fli formula for the generalized Chern-Simons
function on the family of cone-manifold structures of 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 knot orbifolds. For the
fundamental group of 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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