52 research outputs found
An extension of a Bourgain--Lindenstrauss--Milman inequality
Let || . || be a norm on R^n. Averaging || (\eps_1 x_1, ..., \eps_n x_n) ||
over all the 2^n choices of \eps = (\eps_1, ..., \eps_n) in {-1, +1}^n, we
obtain an expression ||| . ||| which is an unconditional norm on R^n.
Bourgain, Lindenstrauss and Milman showed that, for a certain (large)
constant \eta > 1, one may average over (\eta n) (random) choices of \eps and
obtain a norm that is isomorphic to ||| . |||. We show that this is the case
for any \eta > 1
Real eigenvalues in the non-Hermitian Anderson model
The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the
spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity
parameter, are shown to be real and exponentially close to the Hermitian
eigenvalues. This complements previous results, according to which the
eigenvalues in the spectral regions in which the non-Hermiticity parameter
exceeds the Lyapunov exponent are aligned on curves in the complex plane.Comment: 21 pp., 2 fig; to appear in Ann. Appl. Proba
On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices
We apply the methods of classical approximation theory (extreme properties of
polynomials) to study the essential support of the absolutely
continuous spectrum of Jacobi matrices. First, we prove an upper bound on the
measure of which takes into account the value distribution of the
diagonal elements, and implies the bound due to Deift-Simon and
Poltoratski-Remling.
Second, we generalise the differential inequality of Deift-Simon for the
integrated density of states associated with the absolutely continuous spectrum
to general Jacobi matrices.Comment: 18pp, fixed typos (incl. one in title
An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies
We prove an isoperimetric inequality for the uniform measure on a uniformly
convex body and for a class of uniformly log-concave measures (that we
introduce). These inequalities imply (up to universal constants) the
log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric
inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a
concentration inequality for uniformly convex bodies, similar to that proved by
Gromov--Milman.Comment: 39 page
An isoperimetric inequality on the balls
The normalised volume measure on the unit ball ()
satisfies the following isoperimetric inequality: the boundary measure of a set
of measure is at least , where
.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP121 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
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