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Series: Stochastics Seminar

This is survey talk where, both for random words and random permutations, I
will present a panoramic view of the subject ranging from classical results to recent
breakthroughs. Throughout, equivalencies with some directed last passage percolation
models with dependent weights will be pointed out.

Series: Stochastics Seminar

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly
common task, tensor recovery remains a challenging problem because of the delicacy
associated with the decomposition of higher order tensors. We introduce a general
framework of convex regularization for low rank tensor estimation.

Series: Stochastics Seminar

Series: Stochastics Seminar

In the 1970s, Girko made the striking observation that, after centering,
traces of functions of large random matrices have approximately Gaussian
distribution. This convergence is true without any further normalization
provided f is smooth enough, even though the trace involves a number of
terms equal to the dimension of the matrix. This is particularly
interesting, because for some rougher, but still natural observables,
like the number of eigenvalues in an interval, the fluctuations diverge.
I will explain how such results can be obtained, focusing in particular
on controlling the fluctuations when the function is not very regular.

Series: Stochastics Seminar

The Abelian sandpile was invented as a "self-organized critical" model
whose stationary behavior is similar to that of a classical statistical
mechanical system at a critical point. On the d-dimensional lattice,
many variables measuring correlations in the sandpile are expected to
exhibit power-law decay. Among these are various measures of the size of
an avalanche when a grain is added at stationarity: the probability that
a particular site topples in an avalanche, the diameter of an avalanche,
and the number of sites toppled in an avalanche. Various predictions
about these exist in the physics literature, but relatively little is
known rigorously. We provide some power-law upper and lower bounds for
these avalanche size variables and a new approach to the question of
stabilizability in two dimensions.

Series: Stochastics Seminar

Series: Stochastics Seminar

We consider the first-passage percolation model defined on the square lattice
Z^2 with nearest-neighbor edges. The model begins with i.i.d. nonnegative
random variables indexed by the edges. Those random variables can be viewed
as edge lengths or passage times. Denote by T_n the length (i.e. passage
time) of the shortest path from the origin to the boundary of the box
[-n,n] \times [-n,n]. We focus on the case when the distribution function of the
edge weights satisfies F(0) = 1/2. This is sometimes known as the "critical
case" because large clusters of zero-weight edges force T_n to grow at most
logarithmically. We characterize the limit behavior of T_n under conditions
on the distribution function F. The main tool involves a new relation between
first-passage percolation and invasion percolation. This is joint work with
Michael Damron and Wai-Kit Lam.

Series: Stochastics Seminar

In two-dimensional critical percolation, the work of Aizenman-Burchard
implies that macroscopic distances inside percolation clusters are
bounded below by a power of the Euclidean distance greater than 1+\epsilon, for
some positive \epsilon. No more precise lower bound has been given so far.
Conditioned on the existence of an open crossing of a box of side length
n, there is a distinguished open path which can be characterized in
terms of arm exponents: the lowest open path crossing the box. This
clearly gives an upper bound for the shortest path. The lowest crossing
was shown by Morrow and Zhang to have volume n^4/3 on the triangular
lattice.
In 1992, Kesten and Zhang asked how, given the existence of an open
crossing, the length of the shortest open crossing compares to that of
the lowest; in particular, whether the ratio of these lengths tends to
zero in probability. We answer this question positively.

Series: Stochastics Seminar

It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This talk is based on the joint work with G. Paouris and P. Pivovarov.

Series: Stochastics Seminar

For a nonnegative random variable Y with finite nonzero mean
\mu, we say that Y^s has the Y-size bias distribution if E[Yf(Y)] =
\mu E[f(Y^s)] for all bounded, measurable f. If Y can be coupled to
Y^s having the Y-size bias distribution such that for some constant C
we have Y^s \leq Y + C, then Y satisfies a 'Poisson tail' concentration
of measure inequality. This yields concentration results for examples
including urn occupancy statistics for multinomial allocation models and
Germ-Grain models in stochastic geometry, which are members of a class of
models with log concave marginals for which size bias couplings may be
constructed more generally. Similarly, concentration bounds can be shown
when one can construct a bounded zero bias coupling or a Stein pair for a
mean zero random variable Y. These latter couplings can be used to
demonstrate concentration in Hoeffding's permutation and doubly indexed
permutations statistics. The bounds produced, which have their origin in
Stein's method, offer improvements over those obtained by using other
methods available in the literature. This work is joint with J. Bartroff,
S. Ghosh and L. Goldstein.