Columbia Probability Seminar
The seminar covers a wide range of topics in pure and
applied probability. The seminar is organized jointly by the
Mathematics and Statistics departments
and is run by Julien
Dubedat, Clement
Hongler, Fredrik
Johansson Viklund, Ioannis
Karatzas, and Philip Protter.
Email the organizers at
probability_seminar  at  math.columbia.edu.
In the spring, the seminar usually takes place in Statistics SSW Building Room 903 (1255
Amsterdam Avenue between 121st & 122nd Street) on Fridays at 12:00noon1:00pm.
Join the Columbia
Probability Seminar Mailing List.
Last seminar for the Spring Semester: Friday April 22: Gabor Pete
(University of Toronto)
The scaling limits of dynamical and nearcritical percolation, and
the Minimal Spanning Tree
Let each site of the triangular lattice, with small mesh $\eta$,
have an independent Poisson clock with a certain rate $r(\eta) =
\eta^{3/4+o(1)}$ switching between open and closed such that, at any given
moment, the configuration is just critical percolation. In particular, the
probability of a leftright open crossing in the unit square is close to
1/2. Furthermore, because of the scaling, the expected number of switches
in unit time between having a crossing or not is of unit order.
We prove that the limit (as $\eta \to 0$) of the above process exists as a
Markov process, and it is conformally covariant: if we change the domain
with a conformal map $\phi(z)$, then time has to be scaled locally by
$\phi'(z)^{3/4}$. A key step in the proof is that the counting measure
on the set of pivotal points in a percolation configuration has a
conformally covariant scaling limit. Similar limit measures can be
constructed for other special points: length measures on interfaces and
area measures on clusters.
The same proof yields a similar result for nearcritical percolation, and
it also shows that the scaling limit of (a version of) the Minimal
Spanning Tree exists, it is invariant under translations, rotations and
scaling, but *probably* not under general conformal maps. I will explain
what is missing for a proof.
Joint works with Christophe Garban and Oded Schramm.
Spring Semester 2011

Friday, January 28, 2011
 Gregory F. Lawler
(University of Chicago)
Defining the SchrammLoewner evolution in multiply
connected domains with the Brownian loop measure
We define the SchrammLoewner evolution in multiply
connected domains for kappa \leq 4 using the Brownian loop
measure. We show that in the
case of the annulus, this is the same as the measure obtained recently by
Dapeng Zhan. We use the loop formation to give a different derivation of the partial differential equation for the partition function in the annulus.

February 18, 2011
 Johannes MuhleKarbe (ETH Zurich)
Minerva Lectures: Shadow Prices in Portfolio Optimization with Transaction Costs
Time and place:
Lecture I: 46 pm February 1 in Room 312 Math
Lecture II: 46 pm February 3 in
Room 312 Math
Lecture III: 46 pm February 7 in Room 622 Math
Lecture IV: 46 pm February 8 in Room 312 Math.
Abstract and more information on the
Minerva Lectures webpage.
Friday, February 4, 2011
 Clément Hongler (Columbia University)
Ising model interfaces with free boundary conditions
We show conformal invariance of the Ising interfaces in presence of free boundary conditions. In particular we prove the conjecture of Bauer, Bernard and Houdayer about the scaling limit of interfaces arising in a socalled dipolar setup and generalize a result of Chelkak and Smirnov.
The limiting process is a Loewner chain guided by a drifted Brownian motion, known as dipolar SLE or SLE(3,3/2) in the literature. This case is made harder by the absence of natural discrete holomorphic martingales, requiring us to introduce "exotic" martingale observables. The study of these observables is performed by KramersWannier duality and EdwardsSokal coupling, and the computation of the scaling limit is made by appealing to discrete complex analysis methods, to three existing convergence results about discrete fermions, to the scaling limit of critical FortuinKasteleyn model interfaces and to the introduction of Coulomb gas integrals.
Our result allows also to show conjectures by Langlands, Lewis and SaintAubin about conformal invariance of crossing probabilities for the Ising model.
Joint work with Kalle Kytölä.

Friday, February 11, 2011
 Philip Protter
(Columbia University)
Recent Results on Filtration Shrinkage
The expansion of filtrations was a popular subject in the
1980s, as it (among other things) allowed one to increase the space
of possible integrands for stochastic integration. However
filtration shrinkage, its mirror image sister, went largely
unstudied, possibly due to the lack of potential applications.
Current progress in the theory of credit risk, however, has made
filtration shrinkage seem much more important. We study the issue of
the behavior of local martingales under filtration shrinkage (this is
based on joint work with Hans Föllmer), and also the behavior of
compensators of stopping times (based on joint work with with Svante
Janson and Sokhne M'Baye).

Friday, February 18, 2011
 Johannes Ruf
(Columbia University)
Hedging under arbitrage
Explicit formulas for optimal trading strategies in terms of minimal required initial capital are
derived to replicate a given terminal wealth in a continuoustime Markovian context. To achieve
this goal this talk does not assume the existence of an equivalent local martingale measure.
Instead a new measure is constructed under which the dynamics of the stock price processes
simplify. It is shown that delta hedging does not depend on the ``no free lunch with vanishing
risk'' assumption. However, in the case of arbitrage the problem of finding an optimal strategy is
directly linked to the nonuniqueness of the partial differential equation corresponding to the
BlackScholes equation. The recently often discussed phenomenon of ``bubbles'' is a special case
of the setting in this talk.

Friday, February 25, 2011
 Partha Dey
(Courant Institute)
Central Limit Theorem for FirstPassage Percolation along thin cylinders
We consider firstpassage percolation on the graph Z x [h_n, h_n]^{d1} where each edge has an i.i.d. nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the firstpassage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We show that the firstpassage time between the origin and the vertex (n,0,...,0) satisfies a Gaussian CLT as long as h_n << n^{1/(d+1)}. The proof is based on moment estimates, decomposition of the first passage time as an approximate sum of independent random variables and a renormalization argument. We conjecture that the CLT holds upto $h_n=o(n^{2/3})$ for $d=2$ and provide some numerical and heuristic support for that. Based on joint work with Sourav Chatterjee.

Friday, March 4, 2011
 Murad Taqqu
(Boston University and Columbia University)
The Rosenblatt distribution
The Rosenblatt process is the next simplest selfsimilar
process which belongs to the Wiener Chaos after fractional
Brownian motion. While fractional Brownian motion is Gaussian,
the Rosenblatt process is not Gaussian. Thus obtaining the
Rosenblatt distribution which is the distribution of the
Rosenblatt process at time 1 is not easy. Our goal is to derive
various properties of the Rosenblatt distribution and to
describe a technique for computing it numerically with a high
degree of precision. This is joint work with Mark Veillette.

Friday, March 11, 2011
 Spring Recess  No Seminar


Friday, March 18, 2011
 Spring Recess  No Seminar


Friday, March 25, 2011
 Nicolai Krylov
(University of Minnesota)
Filtering partially observable diffusions up to the
exit time from a domain
We consider a two component diffusion
process with the second component treated as
the observations of the first one. The observations
are available only until the first exit time of the first
component from a fixed domain. We derive filtering
equations for the conditional distribution
of the first component before it hits the boundary
and give a formula for the conditional distribution
of the first component at the first time it hits the boundary.

March 29April 1
 Darrell Duffie
(Stanford University)
Minerva Lectures: Dark Markets
Tuesday, March 29
10:30am  12:00pm, SSW 1025
Wednesday, March 30
10:30am  12:00pm, SSW 1025
Friday, April 1
12:00pm  1:00pm, SSW 903
Abstracts and more information on the
Minerva Lectures webpage

Friday, April 1, 2011
Note: two seminars: D. Duffie 12noon1pm,
M. Freidlin 230pm330pm, both in Rm SSW 903.
 Darrell Duffie
(Stanford University) [12noon1pm in Rm SSW 903.]
Dark Markets
The financial crisis of 20072009 brought significant concerns and regulatory action regarding the role of overthecounter markets, particularly from the viewpoint of financial instability. Overthecounter markets for derivatives, collateralized debt obligations, and repurchase agreements played particularly important roles in the crisis and in subsequent legislation in U.S. and Europe. This legislation has also focused on increasing competition and transparency. The modeling of OTC markets, however, is still relatively undeveloped in comparison to the available research on central market mechanisms.
Rather than trading a centralized mechanism such as an auction, specialist, or limitorder book, overthecounter market participants negotiate terms privately with other market participants, often pairwise. Overthecounter investors may be largely unaware of prices that are currently available elsewhere in the market, or of recent transactions prices. In this sense, OTC markets are relatively opaque; investors are somewhat in the dark about the most attractive available terms and about who might offer them. These lectures addresses how prices, asset allocations, and information transmission in OTC markets are influenced by this form of opaqueness. The objective is to provide a brief introduction to OTC markets, including some of the key conceptual issues and modeling techniques, and to provide a foundation for reading more advanced research in this topic area. The lectures assume a graduatelevel background in probability theory.
 Mark Freidlin
(University of Maryland at College Park) [2.30pm3.30pm in Rm SSW 903.]
Stochastic and deterministic perturbations
of systems with multiple invariant measures:
longtime evolution
I will consider perturbations of dynamical systems and stochastic
processes with conservation laws. Longtime behavior of such systems
is defined by slow evolution of first integrals due to the perturbations.
In an appropriate time scale, this slow evolution, even in the case of
pure deterministic perturbations of deterministic systems, should be
approximated, in general, by a stochastic process.
The averaging principle and the large deviation theory are the main
tools in this class of problems.

Friday, April 8, 2011
 Jean Jacod
(Universite Pierre et Marie Curie)
The quadratic variation of an It\^o semimartingale without
Brownian part
In the context of high frequency data, one of the main objects of
interest is the quadratic variation. When the underlying process is an
Itˆo semimartingale one knows the rate of convergence of the
``approximated'' quadratic variation when it is computed on the basis of
a regular sampling with a mesh going to 0. This rate is the square
root of the number of observations.
When the process has no Brownian part, the above limit vanishes,
meaning that the rate is not appropriate. We show that, under some
(unfortunately rather strong) assumptions, there is a faster rate, and
we describe the corresponding limiting process. (Joint work with
Viktor Todorov.)
Abstract in pdf.

Friday, April 15, 2011
 ColumbiaPrinceton Probability Day
(At Princeton University)
Main Speakers:
* Alexei Borodin (Caltech)
* Shige Peng (Shandong U.)
* Yakov Sinai (Princeton)
* HorngTzer Yau (Harvard)
Student Speakers:
* Antonio Auffinger (Courant)
* Hana Kogan (CUNY)
* Ilya Vinogradov (Princeton)
Registration is free and will be open until April 1st, 2011.
The registration form can be found on the Probability Day website. Continental breakfast
and lunch will be provided for all registered participants.
Directions and a schedule can be found on the website. Titles and abstracts
of the talks will be posted there when they are available.

April 16May 3
 Josef Teichmann
(ETH Zurich)
Minerva Lectures

Friday, April 22, 2011
 Gabor Pete
(University of Toronto)
The scaling limits of dynamical and nearcritical percolation, and
the Minimal Spanning Tree
Let each site of the triangular lattice, with small mesh $\eta$,
have an independent Poisson clock with a certain rate $r(\eta) =
\eta^{3/4+o(1)}$ switching between open and closed such that, at any given
moment, the configuration is just critical percolation. In particular, the
probability of a leftright open crossing in the unit square is close to
1/2. Furthermore, because of the scaling, the expected number of switches
in unit time between having a crossing or not is of unit order.
We prove that the limit (as $\eta \to 0$) of the above process exists as a
Markov process, and it is conformally covariant: if we change the domain
with a conformal map $\phi(z)$, then time has to be scaled locally by
$\phi'(z)^{3/4}$. A key step in the proof is that the counting measure
on the set of pivotal points in a percolation configuration has a
conformally covariant scaling limit. Similar limit measures can be
constructed for other special points: length measures on interfaces and
area measures on clusters.
The same proof yields a similar result for nearcritical percolation, and
it also shows that the scaling limit of (a version of) the Minimal
Spanning Tree exists, it is invariant under translations, rotations and
scaling, but *probably* not under general conformal maps. I will explain
what is missing for a proof.
Joint works with Christophe Garban and Oded Schramm.
Fall Semester 2010

Friday, September 24, 2010
 Fredrik Johansson Viklund
(Columbia University)
Convergence rates for looperased random walk
Looperased random walk (LERW) is a selfavoiding random walk obtained by
chronologically erasing the loops of a simple random walk.
In the plane, the lattice size scaling limit of LERW is known to be SLE(2),
a random fractal curve constructed by solving the Loewner
differential equation with a Brownian motion input.
In the talk, we will discuss our recent work on obtaining a rate for the convergence of LERW to SLE(2).
More precisely, we will outline our derivation of a rate for the convergence of the Loewner driving function for looperased random walk
to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2).
We will also, time permitting, discuss how to obtain a rate for the
convergence with respect to Hausdorff distance.
This is joint work with C. Benes (CUNY) and M. Kozdron (U. of Regina).

Friday, October 1, 2010
 Misha Sodin (Tel Aviv University)
Nodal lines of random waves
In the talk, I will introduce random spherical harmonics
and random plane waves, and will describe recent attempts
to understand the mysterious and beautiful structure of
their nodal lines. The talk is based on a joint
work with Fedor Nazarov.

Friday, October 8, 2010
 John Palmer (Arizona)
I will survey some of what is known about the correlations
of the solvable Ising model in two dimensions. This includes
formulas for the lattice correlations, the scaling functions
which exhibit the asymptotics of the lattice correlations at
the correlation length scale, and the short distance behavior
of the scaling functions summarized in correlations predicted
by conformal field theory. Beyond the accident of being
"solvable" there are some surprises in the sort of mathematics
that has applications to understanding the correlations. I
will say something about the role that elliptic curves and
uniformization play in understanding the lattice correlations.

Friday, October 15, 2010
 Henrik Hult (Royal Institute of Technology)
Sanov's theorem under importance sampling
We present a version of Sanov's theorem for the weighted empirical measure based on importance sampling. The theorem provides a way to quantify the probability of large errors of importance sampling estimates. It is, in addition, potentially useful for designing efficient importance sampling algorithms.

Friday, October 22, 2010
 Vadim Gorin (Moscow State University)
From random tilings to representation theory
Lozenge tilings of planar domains provide a simple, yet sophisticated model of random surfaces. Asymptotic behavior of such models has been extensively studied in recent years.
We will start from recent results about qdistributions on tilings of a hexagon or, equivalently, on boxed plane partitions. (This part is based on the joint work with A.Borodin and E.Rains).
In the second part of the talk we will explain how representation theory of the infinitedimensional unitary group is related to random lozenge tilings with a certain Gibbs property. We will discuss applications of this correspondence and results on the classification of Gibbs measures on tilings of the halfplane.

Friday, October 29, 2010
 Richard Bass (University of Connecticut)
Convergence of symmetric Markov chains
Suppose one has a symmetric Markov chain on the
integer lattice in one or more dimensions. One might hope that, properly
normalized, in the limit one would obtain a diffusion that corresponds to a
second order elliptic operator in divergence form. I will give a
sufficient condition for such a result to hold, even when the
symmetric Markov chain is not a nearest neighbor chain or one
with bounded range. Then I will discuss generalizations where the limit
process has jumps as well as a continuous part. This is joint work
with Takashi Kumagai and Toshihiro Uemura.

Friday, November 5, 2010
 Ivan Corwin (Courant Institute)
The KPZ universality class and equation
The Gaussian central limit theorem says that for a wide
class of stochastic systems, the bell curve (Gaussian distribution)
describes the statistics for random fluctuations of important
observables. In this talk I will look beyond this class of systems to
a collection of probabilistic models which include random growth
models, polymers, particle systems, matrices and stochastic PDEs, as
well as certain asymptotic problems in combinatorics and
representation theory. I will explain in what ways these different
examples all fall into a single new universality class with a much
richer mathematical structure than that of the Gaussian.

Wednesday, November 10, 2010, 56 PM, Math 520. Special joint Probability/Geometry/Analysis Seminar.
 HorngTzer Yau (Harvard University)
Universality of random matrices and Dyson Brownian motion
Random matrices were introduced by E. Wigner to model the excitation
spectra of large nuclei.
The central idea is based on the hypothesis that the local statistics
of the excitation spectrum for a large complicated system is
universal
in the sense that it depends only on the symmetry class of the
physical system but not on other detailed structures.
Dyson Brownian motion is the flow of eigenvalues of random matrices
when each matrix element performs independent Brownian motions.
In this lecture, we will explain the connection between the
universality of random matrices and
the approach to local equilibrium of Dyson Brownian motion. This
connection has led to a complete solution
of the universality conjecture by Wigner, Dyson and Mehta.
The main tools in our approach are an estimate on the flow of entropy
in Dyson Brownian motion and a local semicircle law.
One key feature of the entropy estimate is an extension of the
logarithmic Sobolev inequality
to cases not covered by the convexity criterion of Bakry and Emery.

Friday, November 12, 2010
 Jason Miller (Stanford University)
CLE(4) and the Gaussian Free Field
The discrete Gaussian free field (DGFF) is the Gaussian measure on
functions $h \colon D \to \R$, $D \subseteq \Z^2$ bounded, with
covariance given by the Green's function for simple random walk. The
graph of $h$ is a random surface which serves as a physical model for
an effective interface. We show that the collection of random loops
given by the level sets of the DGFF for any height $\mu \in \R$
converges in the finemesh scaling limit to a family of loops which is
invariant under conformal transformations when $D$ is a lattice
approximation of a nontrivial simply connected domain. In
particular, there exists $\lambda > 0$ such that the level sets whose
height is an odd integer multiple of $\lambda$ converges to a nested
conformal loop ensemble with parameter $\kappa=4$ [socalled
$\CLE(4)]$, a conformally invariant measure on loops which locally
look like $\SLE(4)$. Using this result, we give a coupling of the
continuum Gaussian free field (GFF), the finemesh scaling limit of
the DGFF, and $\CLE(4)$ such that the GFF can be realized as a
functional of $\CLE(4)$ and conversely $\CLE(4)$ can be made sense as
a functional of the GFF. This is joint work with Scott Sheffield.

ThursdayFriday, November 1819, 2010
 Ninth Northeast Probability Seminar (At CUNY Graduate Center)
Speakers:
Nathanael Berestycki (Cambridge University)
Persi Diaconis (Stanford University)
Yves Le Jan (Universite Paris Sud)
Edwin Perkins (University of British Columbia)
Further information

Friday, November 26, 2010

Friday, December 3, 2010
 Boris Rozovsky (Brown University)
Stochastic Quantization and NavierStokes Equation
Minerva Lectures

Friday, December 10, 2010
 Grigori Olshanski (Dobrushin Math. Lab., Kharkevich Institute, Moscow;
Independent University of Moscow)
A model of infinitedimensional Markov dynamics
The talk is based on recent joint work with Alexei Borodin. The problem
of harmonic analysis on the infinitedimensional unitary group
U(infinity) leads to a family of probability measures living on an
infinitedimensional space Omega, a kind of dual object to
U(infinity). The measures in question can also be interpreted as
determinantal point processes; earlier we explicitly computed their
correlation functions. Now we construct continuoustime Markov processes
on Omega, preserving those measures. In contrast to many works on
infinitedimensional Markov dynamics, our approach does not use
Dirichlet forms or other advanced analytic techniques, it relies on a
mix of elementary tools from Markov chains and some ideas of random
matrix theory.
Spring Semester 2010
Please see Spring
Semester 2010 and older.