- You are here:
- GT Home
- Home
- News & Events

Series: Stochastics Seminar

We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. This is joint work with Marek Biskup.

Series: Stochastics Seminar

The Eden model, a special case of first-passage percolation, is a
stochastic growth model in which an infection that initially occupies the
origin of Z^d spreads to neighboring sites at rate 1. Infected sites are
colonized permanently; that is, an infected site never heals. It is known
that at time t, the infection occupies a set B(t) of vertices with volume of
order t^d, and the rescaled set B(t)/t converges to a convex, compact
limiting shape. In joint work with J. Hanson and W.-K. Lam, we partially
answer a question of K. Burdzy, concerning the order of the size of the
boundary of B(t). We show that, in various senses, the boundary is
relatively smooth, being typically of order t^{d-1}. This is in contrast to
the fractal behavior of interfaces characteristic of percolation models.

Series: Stochastics Seminar

Form a multiset by including Poisson(1/k) copies of each
positive integer k, and consider the sumset---the set of all finite sums
from the Poisson multiset. It was shown recently that four such
(independent) sumsets have a finite intersection, while three have
infinitely many common elements. Uncoincidentally, four uniformly random
permutations will invariably generate S_n with asymptotically positive
probability, while three will not. What is so special about four? Not much.
We show that this result is a special case of the "ubiqituous" Ewens
sampling formula. By varying the distribution's parameter we can vary the
number of random permutations needed to invariably generate S_n, and,
relatedly, the number of Poisson sumsets to have finite intersection.
*Joint with Gerandy Brita Montes de Oca, Christopher Fowler, and Avi Levy.

Series: Stochastics Seminar

The bootstrap procedure is well known for its good finite-sample performance, though the
majority of the present results about its accuracy are asymptotic. I will study the
accuracy of the
weighted (or multiplier) bootstrap procedure for estimation of quantiles of a likelihood
ratio statistic.
The set-up is the following: the sample size is bounded, random observations are
independent,
but not necessarily identically distributed, and a parametric model
can be misspecified. This problem had been considered in the recent work of Spokoiny and
Zhilova (2015)
with non-optimal results. I will present a new approach improving the existing results.

Series: Stochastics Seminar

We look at a class of Hermitian random matrices which includes Wigner
matrices, heavy-tailed random matrices, and sparse random matrices such as
adjacency matrices of Erdos-Renyi graphs with p=1/N. Our matrices have real
entries which are i.i.d. up to symmetry. The distribution of entries
depends on N, and we require sums of rows to converge in distribution; it
is then well-known that the limit must be infinitely divisible.
We show that a limiting empirical spectral distribution (LSD) exists, and
via local weak convergence of associated graphs, the LSD corresponds to the
spectral measure associated to the root of a graph which is formed by
connecting infinitely many Poisson weighted infinite trees using a backbone
structure of special edges. One example covered are matrices with i.i.d.
entries having infinite second moments, but normalized to be in the
Gaussian domain of attraction. In this case, the LSD is a semi-circle law.

Series: Stochastics Seminar

Motivated by problems in turbulent mixing, we consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups. We study the ergodic properties and provide criteria that ensure the Hormander condition for the corresponding Markov processes on phase space. Two different types of models are considered: the first one is a classical Langevin type perturbation and the second one is a perturbation by a “conservative noise”. We also study an example of a non-compact group. Joint work with Vladimir Sverak.

Series: Stochastics Seminar

Given a simple connected graph G=(V,E), the abelian sandpile
Markov chain evolves by adding chips to random vertices and then
stabilizing according to certain toppling rules. The recurrent states form
an abelian group \Gamma, the sandpile group of G. I will discuss joint
work with Dan Jerison and Lionel Levine in which we characterize the
eigenvalues and eigenfunctions of the chain restricted to \Gamma in terms
of "multiplicative harmonic functions'' on V. We show that the moduli of
the eigenvalues are determined up to a constant factor by the lengths of
vectors in an appropriate dual Laplacian lattice and use this observation
to bound the mixing time of the sandpile chain in terms of the number of
vertices and maximum vertex degree of G. We also derive a surprising
inverse relationship between the spectral gap of the sandpile chain and
that of simple random walk on G.

Series: Stochastics Seminar

The focus of my talk will be stochastic forms of isoperimetric
inequalities for convex sets. I will review some fundamental
inequalities including the classical isoperimetric inequality and
those of Brunn-Minkowski and Blaschke-Santalo on the product of
volumes of a convex body and its polar dual. I will show how one can
view these as global inequalities that arise via random approximation
procedures in which stochastic dominance holds at each stage. By laws
of large numbers, these randomized versions recover the classical
inequalities. I will discuss when such stochastic dominance arises
and its applications in convex geometry and probability. The talk
will be expository and based on several joint works with G. Paouris,
D. Cordero-Erausquin, M. Fradelizi, S. Dann and G. Livshyts.

Series: Stochastics Seminar

In the late 80's, several relationships have been established
between the Information Theory and Convex Geometry, notably
through the pioneering work of Costa, Cover, Dembo and Thomas.
In this talk, we will focus on one particular relationship. More
precisely, we will focus on the following conjecture of Bobkov,
Madiman, and Wang (2011), seen as the analogue of the
monotonicity of entropy in the Brunn-Minkowski theory:
The inequality
$$ |A_1 + \cdots + A_k|^{1/n} \geq \frac{1}{k-1} \sum_{i=1}^k
|\sum_{j \in \{1, \dots, k\} \setminus \{i\}} A_j |^{1/n}, $$
holds for every compact sets $A_1, \dots, A_k \subset
\mathbb{R}^n$. Here, $|\cdot|$ denotes Lebesgue measure in
$\mathbb{R}^n$ and $A + B = \{a+b : a \in A, b \in B \}$ denotes
the Minkowski sum of $A$ and $B$.
(Based on a joint work with M. Fradelizi, M. Madiman, and A.
Zvavitch.)

Series: Stochastics Seminar

Paper available on arXiv:1412.3661

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities Pr(n−1/2∑ni=1Xi∈A) where X1,…,Xn are independent random vectors in ℝp and Ais a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn→∞ as n→∞ and p≫n; in particular, p can be as large as O(eCnc) for some constants c,C>0. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.