146 research outputs found
On the spectrum of sum and product of non-hermitian random matrices
In this short note, we revisit the work of T. Tao and V. Vu on large
non-hermitian random matrices with independent and identically distributed
entries with mean zero and unit variance. We prove under weaker assumptions
that the limit spectral distribution of sum and product of non-hermitian random
matrices is universal. As a byproduct, we show that the generalized eigenvalues
distribution of two independent matrices converges almost surely to the uniform
measure on the Riemann sphere.Comment: 8 pages, statement of main theorem slightly improve
A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts
It was conjectured by Alon and proved by Friedman that a random -regular
graph has nearly the largest possible spectral gap, more precisely, the largest
absolute value of the non-trivial eigenvalues of its adjacency matrix is at
most with probability tending to one as the size of the
graph tends to infinity. We give a new proof of this statement. We also study
related questions on random -lifts of graphs and improve a recent result by
Friedman and Kohler.Comment: 49 pages, final version, to appear in "Annales scientifiques de
l'\'Ecole normale sup\'erieure
Eigenvalues of Euclidean Random Matrices
We study the spectral measure of large Euclidean random matrices. The entries
of these matrices are determined by the relative position of random points
in a compact set of . Under various assumptions we establish
the almost sure convergence of the limiting spectral measure as the number of
points goes to infinity. The moments of the limiting distribution are computed,
and we prove that the limit of this limiting distribution as the density of
points goes to infinity has a nice expression. We apply our results to the
adjacency matrix of the geometric graph.Comment: 16 pages, 1 figur
Navigation on a Poisson point process
On a locally finite point set, a navigation defines a path through the point
set from one point to another. The set of paths leading to a given point
defines a tree known as the navigation tree. In this article, we analyze the
properties of the navigation tree when the point set is a Poisson point process
on . We examine the local weak convergence of the navigation
tree, the asymptotic average of a functional along a path, the shape of the
navigation tree and its topological ends. We illustrate our work in the
small-world graphs where new results are established.Comment: Published in at http://dx.doi.org/10.1214/07-AAP472 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Combinatorial optimization over two random point sets
We analyze combinatorial optimization problems over a pair of random point
sets of equal cardinal. Typical examples include the matching of minimal
length, the traveling salesperson tour constrained to alternate between points
of each set, or the connected bipartite r-regular graph of minimal length. As
the cardinal of the sets goes to infinity, we investigate the convergence of
such bipartite functionals.Comment: 34 page
Around the circular law
These expository notes are centered around the circular law theorem, which
states that the empirical spectral distribution of a nxn random matrix with
i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the
complex plane as the dimension tends to infinity. This phenomenon is the
non-Hermitian counterpart of the semi circular limit for Wigner random
Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random
covariance matrices. We present a proof in a Gaussian case, due to Silverstein,
based on a formula by Ginibre, and a proof of the universal case by revisiting
the approach of Tao and Vu, based on the Hermitization of Girko, the
logarithmic potential, and the control of the small singular values. Beyond the
finite variance model, we also consider the case where the entries have heavy
tails, by using the objective method of Aldous and Steele borrowed from
randomized combinatorial optimization. The limiting law is then no longer the
circular law and is related to the Poisson weighted infinite tree. We provide a
weak control of the smallest singular value under weak assumptions, using
asymptotic geometric analysis tools. We also develop a quaternionic
Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment
- âŠ