Probability Seminar --- 2005 -- 2006
Sept. 5 --- Greg Lawler
Conformal Invariance and Two-dimensional Polymers
This will be a "colloquium level" talk reviewing work in the last
few years on the scaling limit of self-avoiding walks. I will be
giving this talk at a meeting in Pittsburgh later in the week, so I
thought I could start the semester by giving it here.
Sept. 12 --- Joan Lind, The geometry of Loewner evolution
Sept. 19 --- David White, Processes with inert drift
Sept. 26 --- Josh Rushton, Cornell,
A Chung-type functional LIL for $\alpha$-stable processes, and related
invariance results.
Oct. 3 --- Paul Jung, Cornell
Oct. 10 --- break
Oct. 17 --- Lea Popovic, Continuum trees from catalytic population models
Oct 24 --- Raazesh Sainudiin,
A randomized enclosure algorithm: Moore rejection sampling
Oct. 31 Kazumasa Kuwada, Large deviations for random currents
induced from stochastic line integrals
Nov, 7 Gady Kozma, The scaling limit of loop-erased
walk in three dimensions
Nov, 11 (Special time), 11:00 am, Julien Dubedat
Nov. 21 --- Rick Durrett,
Waiting for ATCAAAG
One possible explanation for the substantial organismal differences that
have developed in the 6 million years sincee the divergence of humans and
chimpanzees is that there have been changes in gene regulation. The word in
the title is a sample transcription factor binding sites and motivates the
following probability question: given a 1000 nucleotide region in our
genome, how long does it take for a specified six to nine letter word to
appear in that region in some individual? Stone and Wray (2001) computed
5949 years as the answer for six letter words in the human population. We
will show that for words of length 6, the average waiting time is 100,000
years while for words of length 8, the waiting time is roughly a 1/3 - 2/3
mixture of exponentials with means 375,000 years and 625 million years. In
biological reality, the match to the target word does not have to be
perfect for binding to occur. If we model this by saying that we allow
mismatch, then almost all of the mass in the probability distribution
shifts to the smaller mean. This is joint work with Deena Schmidt.
SPRING 2006 TENTATIVE SCHEDULE
Jan 23 --- N. Lanchier
Jan 30 --- Two talks: S. Cohen and A. Sturm (details later)
Feb 6 -- no talk, but see
http://www.cs.cornell.edu/courses/cs789/2006sp/Ludek.Kucera.htm
for an interesting, related talk.
Feb 13 -- Antar Bandyopadhyay, Chalmers University,
Random Walk in Dynamic Markovian Random Environment
Thursday, Feb 16, 3:00, 206 Malott (note unusual time and place)
Davar Khoshnevisan, Utah
A Coupling and the Darling - Erdos Conjectures
We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process
with an i.i.d. sequence.
We then apply this coupling to resolve two conjectures of Darling and
Erds (1956).
Interestingly enough, we prove one and disprove the other conjecture.
[This is joint work with David Levin.]
Time-permitting, we may use the ideas of this talk to describe
precisely the rate of convergence in the classical law of the
iterated logarithm of Khintchine for Brownian motion (1933).
[This portion is joint work with David Levin and Zhan Shi, and has
recently appeared in
the Electr. Comm. of Probab. (2005)]
Feb. 27 -- Mihai Sirbu, Columbia
Mar 6 -- Ted Cox, Syracuse
Mar. 13 -- X. Chen, Tennessee
Large deviations for Brownian intersection local times and
related problems
It has been known that the study of intersection local times is partially
motivated by the needs in mathematical physics.
In this talk I will report some recent progress on the large deviations
for the intersection local times run by Brownian motions. In particular,
I will give answer to a conjecture raised by the physicist Dulpantier.
In addition, I will talk about some impact of our results in some related
areas such as ranges of random walks and local times of
multi-parameter processes.
Part of the talk is based on the joint work with Richard Bass,
Wenbo Li and Jay Rosen.
Mar. 27 -- J. Schweinsburg, UCSD
The loop-erased random walk and the uniform
spanning tree on the four-dimensional discrete torus.
Let x and y be points chosen uniformly at random from the
four-dimensional discrete torus with side length n. We show that the
length of the loop-erased random walk from x to y is of order
n^2 (log n)^{1/6}, resolving a conjecture of Benjamini and Kozma.
We also show that the scaling limit of the uniform spanning tree on
the four-dimensional discrete torus is the Brownian continuum random
tree of Aldous. Our proofs use the techniques developed by Peres and
Revelle, who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the d-dimensional
discrete torus for d >= 5, in combination with results of Lawler
concerning intersections of four-dimensional random walks
Apr. 3, Benedek Valko, U of Toronto,
Limits of random trees from real-world networks
The following random tree model is often used to describe
real-world networks. We start with a single vertex and in every step we
connect a new vertex randomly to one of the old ones with probability
proportional to a function of its degree. We find asymptotic degree
distribution, and prove a limit theorem for the tree itself as viewed from
a random point. (Joint work with A. Rudas and B. Toth)
Apr. 10 -- M. Reed, Duke
Apr. 17 -- J. Yukich, Lehigh
Apr, 24 -- M. Roeckner, Purdue
"Stochastic porous media and fast diffusion equations"
We first present a new existence and uniqueness result for stochastic
evolution equations on Hilbert spaces. This is a generalization of a
classical result by Krylov and Rozovskii based on the so-called
variational approach to stochastic partial differential equations (SPDE).
The main motivation are applications to nonlinear SPDE of porous media
type which also include cases where the nonlinear functions grow slowly at
infinity ("fast diffusion equations"). Generally, the main problem is
to find the appropriate Gelfand triple to work on. In our case Orlics
spaces turn out to be convenient. We show how one must choose the
defining Young function for a given nonlinearity. After presenting these
applications, we shall summarize results about the qualitative behaviour
of solutions and about their invariant measures.