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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.

Series: Stochastics Seminar

In this talk, we propose moment identities for point processes. After
revisiting the case of Poisson point processes, we propose a direct approach to
derive (joint factorial) moment identities for point processes admitting Papangelou
intensities. Applications of such identities are given to random transformations of
point processes and to their distribution invariance properties.

Series: Stochastics Seminar

Motivated by understanding the intricate combinatorial structure
of the Poisson chaos in order to see whether or not a fourth moment type
theorem may hold on that space, we define, construct and study the free
Poisson chaos, a non-commutative counterpart of the classical Poisson
space, on which we prove the free counter part of the fourth moment
theorem. This is joint work with Giovanni Peccati.

Series: Stochastics Seminar

I will discuss the limit theorems for composition of analytic functions on the upper-half-plane, and the analogies and differences with the limit theorems for sums of independent random variables. The analogies are enhanced by recalling that the probabilistic limit theorems are really results about convolution of probability measures, and by introducing a new binary operation on probability measures, the monotone convolution.This is joint work with John D. Williams.

Series: Stochastics Seminar

Ergodic theory of randomly forced space-time homogeneous Burgers
equation in noncompact setting has been developed in a recent paper by
Eric Cator , Kostya Khanin, and myself. The analysis is based on first
passage percolation methods that allow to study coalescing one-sided
action minimizers and construct the global solution via Busemann
functions. i will talk about this theory and its extension to the case
of space-continuous kick forcing. In this setting, the minimizers do
not coalesce, so for the ergodic program to go through, one must use
new soft results on their behavior to define generalized Busemann
functions along appropriate subsequences.

Series: Stochastics Seminar

We define the class of ultra sub-Gaussian random vectors and
derive optimal comparison of even moments of linear combinations of such
vectors in the case of the Euclidean norm. In particular, we get optimal
constants in the classical Khinchine inequality. This is a joint work with
Krzysztof Oleszkiewicz.