Columbia Probability Seminar
Spring Semester 2010
Tuesday, January 26, 10-11am, Math 507 [note special day, time, location]
Clément
Hongler,
U.
Genève
The energy
density in the 2D Ising model
Friday, January 29
Amir
Yehudayoff, IAS
Pseudorandomness
for
finite
groups
Friday, February 5
Olivier
Bernardi,
MIT
From random
trees to random surfaces
Friday, February 12
Xuedong
He,
Columbia
IEOR
Hope, Fear and
Aspiration
Friday, February 19
Kay
Kirkpatrick, Courant Institute
Bose-Einstein
condensation
and
the
nonlinear
Schrodinger equation
Friday,
February
26:
CANCELLED
Friday,
March
5
Stilian
Stoev,
U.
Michigan
Stats
Max-stable
processes: ergodicity, classification, and some new results on their
path properties
March 22-25: Minerva Lectures
Walter
Schachermayer,
Vienna
The
asymptotic theory of transaction costs
School of Social Work 903
Mon Mar 22: 4-5:30pm
Tue Mar 23: 3-4:30pm
Thu Mar 25: 3-4:30pm
March
25-
April
2:
Minerva Lectures
Geoffrey
Grimmett,
Cambridge
Lattice
models
in probability
Thu Mar 25: 9:20-10:50 am, Math 203
Fri Mar 26: 9:20-10:50am, Math 203
Mon Mar 29: 3-4:30 pm, Math 520 (note
time, location)
Tue Mar 30: 9:20-10:50am, Math 203
Thu Apr 1: 9:20-10:50am, Math 203
Friday, April 9
William
Sudderth,
U.
Minnesota
Stats
De Finetti coherence
and logical consistency
April 12-
22: Minerva Lectures
Ofer Zeitouni, UMN & Weizmann Institute
Random
Walks in Random Environments
Mon Apr 12
Fri Apr 16
Mon Apr 19
Wed Apr 21
Thu Apr 22
9:20-10:50 am, Math 622
Friday, April 23
Van
Vu,
Rutgers
Random matrices:
Universality of the Local eigenvalues statistics
Friday, April 30
Steven
Shreve,
Carnegie
Mellon
U.
Diffusion
matching of statistics of an Ito process
Jan 26:
Clément Hongler, U.
Genève
The energy density
in the 2D Ising model
We consider the critical planar
Ising model from a conformal invariance
point of view, using discrete complex analysis techniques. We are
interested in the scaling limit of the model in a simply connected
domain with pure (+ or free) boundary conditions.
More precisely, we will be interested in the behaviour of the energy
density field in the scaling limit, proving an improved version of
conjectures coming from Conformal Field Theory, exhibiting a nice
connection with hyperbolic geometry.
The derivation is made by using a fermionic observable, which is a
discrete holomorphic deformation of a partition function, and then by
passing to the limit, obtaining a continuous holomorphic function,
which solves a certain boundary value problem.
Joint work with Stas Smirnov.
Jan 29: Amir Yehudayoff, IAS
Pseudorandomness
for
finite
groups
We will discuss pseudorandomness with respect to a natural
family of
tests defined by finite groups. For every n and a finite group G,
consider the following family of tests T_n(G): a function t from
{0,1}^n to G in T_n(G) is defined by choosing g_1,...,g_n from G, and
setting t(x_1,...,x_n) = g_1^{x_1} \cdots g_n^{x_n}.
Our goal is to construct a distribution D on {0,1}^n with as small
support as possible
that
epsilon-fools groups of size g. Namely, so that for every group G of
size at most g, and for every test t in
T_n(G), the statistical distance between t(U_n) and t(D) is at most
epsilon, where U_n in the uniform distribution on {0,1}^n. In other
words, no test defined by a group of size at most g can distinguish D
from the uniform distribution with probability higher than epsilon.
We construct a distribution D that epsilon-fools groups of
size g so that log(support(D)) is order log(n)(loglog(n) +
log(1/epsilon) + log(1/g)).
Joint work with Mark Braverman, Anup Rao and Ran Raz.
Feb 5:
Olivier
Bernardi, MIT
From
random trees to random surfaces
Imagine gluing some polygons together by identifying edges in pairs.
If the surface obtained is (homeomorphic to) the sphere, then your
gluing is called a ``planar map''. Equivalently, a planar map can be
defined as a connected planar graph embedded in the sphere (considered
up to homeomorphism).
Maps are of interest, in particular, for defining random geometries.
Indeed, by considering all possible ways of obtaining the sphere by
gluing n squares, one obtains a ``random lattice''. By considering the
continuous limit of random lattices one obtains a ``random surface''.
Many recent advances in understanding these random geometries are
based on bijections between planar maps and certain decorated plane
trees. Of particular importance is a bijection by Schaeffer and its
generalization by Bouttier, Di Francesco and Guitter.
In this talk, I will present a bijection between "tree-rooted
maps"
(planar map with a marked spanning tree) and pairs of plane trees.
Then, I will present the bijection of Schaeffer et al., which can be
seen as a specialization of the previous one. Lastly, (in order boost
the probabilistic content of the talk) I will briefly review some of
the known and unknown properties of random surfaces.
Feb 12: Xuedong He, Columbia IEOR
Hope, Fear
and Aspiration
In this paper, we propose a new portfolio choice model in continuous
time
which features three key human incentives in choice-making: hope, fear
and
aspiration. By applying recently developed quantile formulation, we
solve
this model completely. Three quantitative indices: fear index, hope
index
and lottery-likeness index are proposed to study the impact of hope,
fear
and aspiration respectively on the investment behavior. We find that the
extreme fear would prevent the agent from risking too much and
consequently
induces a portfolio insurance policy endogenously. On the other side,
the
hope will drive the agent aggressive, and the more hopeful he is, the
more
aggressive he will be. Finally, a high aspiration will lead to a
lottery-like terminal payoff, indicating that the agent will risk much.
The
higher the aspiration is, the more risk the agent would or have to take.
This is joint work with Xunyu Zhou.
Feb 19: Kay Kirkpatrick, Courant Institute
Bose-Einstein
condensation
and
the
nonlinear
Schrodinger equation
Near absolute zero, a gas of quantum particles can condense into an
unusual state of matter, called Bose-Einstein condensation, that
behaves like a giant quantum particle. It's only recently that we've
been able to make the rigorous probabilistic connection between the
physics of the microscopic dynamics and the mathematics of the
macroscopic model, the cubic nonlinear Schrodinger equation (NLS).
I'll discuss joint work with Benjamin Schlein and Gigliola Staffilani
on two-dimensional cases for Bose-Einstein condensation--and the
periodic case is especially interesting, because of techniques from
analytic number theory and applications to quantum computing. As time
permits, I'll also mention work in progress on a high-probability
phase transition for the invariant measures of the NLS.
Mar 5: Stilian
Stoev, U. Michigan Stats
Max-stable
processes:
ergodicity,
classification,
and
some new results on their
path properties
We start by reviewing the spectral representations of
max-stable processes. We then discuss results on their ergodicity, weak
mixing, mixing, and classification. We conclude with some work in
progress
on limit theorems in Holder spaces, which can be applied to establish
the
regularity of some Brown-Resnick processes.
Apr
23: Van Vu, Rutgers
Random
matrices:
Universality
of
the
Local
eigenvalues statistics
One
of the main goals of the theory of random matrices is to establish the
limiting distributions of the eigenvalues. In the 1950s, Wigner proved
his famous semi-cirle law (subsequently extended by Anord, Pastur and
others), which established the global distribution of the eigenvalues
of random Hermitian matrices. In the last fifty years or so, the focus
of the theory has been on the local distributions, such as the
distribution of the gaps between consecutive eigenvalues, the k-point
correlations, the local fluctuation of a particular eigenvalue, or the
distribution of the least singular value. Many of these problems have
connections to other fields of mathematics, such as combinatorics,
number theory, statistics and numerical linear algebra.
Most
of the local statistics can be computed explicitly for random matrices
with gaussian entries (GUE or GOE), thanks to Ginibre's formulae of the
joint density of eigenvalues. It has been conjectured that these
results can be extended to other models of random matrices. This is
generally known as the Universality phenomenon, with several specific
conjectures posed by Wigner, Dyson, Mehta etc.
In
this talk, we would like to discuss recent progresses concerning the
Universality phenomenon, focusing on a recent result (obtained jointly
with T. Tao), which asserts that all local statistics of eigenvalues of
a random matrix are determined by the first four moments of the
entries. This (combining with results of Johansson,
Erdos-Ramirez-Schlein-Yau and many others) provides the answer to
several old problems.
The method also extends to other models of random matrices, such as
sample covariance matrices.
Apr 30: Steven
Shreve,
Carnegie
Mellon
U.
Diffusion matching of statistics of an Ito
process
Suppose we are given a multi-dimensional Ito process, which can be
regarded as a model for an underlying asset price together with related
stochastic processes, e.g., volatility. The drift and diffusion
terms for this Ito process are permitted to be arbitrary adapted
processes. We construct a weak solution to a diffusion-type
equation that matches the distribution of the Ito process at each fixed
time. Moreover, we show how to also match the distribution at
each fixed time of statistics of the Ito process, including the running
maximum and running average of one of the components of the process.
A consequence of this result is that a wide variety of exotic
derivative securities have the same prices when written on the original
Ito process or on the mimicking process. This is joint work with
Gerard Brunick.