125 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

### 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 $\mathbb{R}^d$. 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

### 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 $n$ 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

### Large deviations of empirical neighborhood distribution in sparse random graphs

Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is
present independently with probability c/n, with c>0 fixed. For large n, a
typical random graph locally behaves like a Galton-Watson tree with Poisson
offspring distribution with mean c. Here, we study large deviations from this
typical behavior within the framework of the local weak convergence of finite
graph sequences. The associated rate function is expressed in terms of an
entropy functional on unimodular measures and takes finite values only at
measures supported on trees. We also establish large deviations for other
commonly studied random graph ensembles such as the uniform random graph with
given number of edges growing linearly with the number of vertices, or the
uniform random graph with given degree sequence. To prove our results, we
introduce a new configuration model which allows one to sample uniform random
graphs with a given neighborhood distribution, provided the latter is supported
on trees. We also introduce a new class of unimodular random trees, which
generalizes the usual Galton Watson tree with given degree distribution to the
case of neighborhoods of arbitrary finite depth. These generalized Galton
Watson trees turn out to be useful in the analysis of unimodular random trees
and may be considered to be of interest in their own right.Comment: 58 pages, 5 figure

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