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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.

Series: Stochastics Seminar

Log Brunn-Minkowski conjecture was proposed by Boroczky, Lutwak,
Yang and Zhang in 2013. It states that in the case of symmetric convex sets
the classical Brunn-MInkowski inequality may be improved. The Gaussian
Brunn-MInkowski inequality was proposed by Gardner and Zvavitch in 2007. It
states that for the standard Gaussian measure an inequality analogous to the
additive form of Brunn_minkowski inequality holds true for symmetric convex
sets. In this talk we shall discuss a derivation of an equivalent
infinitesimal versions of these inequalities for rotation invariant measures
and a few partial results related to both of them as well as to the
classical Alexander-Fenchel inequality.

Series: Stochastics Seminar

Series: Stochastics Seminar

For a random (complex) entire function, what can we say about the
behavior of the zero set of its N-th derivative, as N goes to infinity?
In this talk, we shall discuss the result of repeatedly differentiating a
certain class of random entire functions whose zeros are the points of a
Poisson process of intensity 1 on the real line. We shall also discuss the
asymptotic behavior of the coefficients of these entire functions. Based on
joint work with Robin Pemantle.

Series: Stochastics Seminar

We prove a central limit theorem for a class of additive processes that
arise naturally in the theory of finite horizon Markov decision problems.
The main theorem generalizes a classic result of Dobrushin (1956) for
temporally non-homogeneous Markov chains, and the principal innovation is
that here the summands are permitted to depend on both the current state
and a bounded number of future states of the chain. We show through several
examples that this added flexibility gives one a direct path to asymptotic
normality of the optimal total reward of finite horizon Markov decision
problems. The same examples also explain why such results are not easily
obtained by alternative Markovian techniques such as enlargement of the
state space. (Joint work with J. M. Steele.)