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Probability 2008-2009
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Fall 2009
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September 11
Ofer Zeitouni UMN and Weizmann Institute
The single ring theorem
Consider an n-dimensional
real
diagonal matrix D, with the empirical measure of the entries close
to some measure P.
Form the matrix X=UDV where U and V are random unitaries.
How does the empirical measure of the (complex) eigenvalues of
X looks like? Surprisingly, even if the support of P consists of several
intervals, the empirical measure of X is supported (asymptotically)
on a single ring. This was proved by physicists Feinberg and Zee.
I will explain the background and a recent proof by
Guionnet, Krishnapur and myself.
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September 18
No seminar this week, due to
Conference in honor of K. Athreya Iowa State University
Happy Jewish New Year
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September 25
Maury Bramson UMN
Positive Recurrence of Reflecting Brownian Motion in 3 Dimensions
A substantial amount of effort has been devoted to studying
semimartingale reflecting Brownian motions (SRBMs) in queueing theory, where
they are often interpreted as limiting idealizations of queueing networks
that are easier to analyze than the original models. Precise conditions are
known for positive recurrence of SRBM in 2 dimensions. The behavior in 4 and
more dimensions is largely unexplored. In 3 dimensions, sufficient
conditions for positive recurrence are given in a paper by El Kharroubi et
al. (2002). In joint work with Jim Dai and Mike Harrison, it has recently
been shown that these results are, in fact, necessary. In this talk, we
summarize the above material.
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October 2
Maury Bramson UMN
Positive Recurrence of Reflecting Brownian Motion in 3 Dimensions (part II)
A substantial amount of effort has been devoted to studying
semimartingale reflecting Brownian motions (SRBMs) in queueing theory, where
they are often interpreted as limiting idealizations of queueing networks
that are easier to analyze than the original models. Precise conditions are
known for positive recurrence of SRBM in 2 dimensions. The behavior in 4 and
more dimensions is largely unexplored. In 3 dimensions, sufficient
conditions for positive recurrence are given in a paper by El Kharroubi et
al. (2002). In joint work with Jim Dai and Mike Harrison, it has recently
been shown that these results are, in fact, necessary. In this talk, we
summarize the above material.
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October 9
Stefano Olla Ceremade
From microscopic dynamics to heat equation: a weak coupling approach
We consider a chain of weakly coupled oscillators whose
Hamiltonian dynamics is perturbed by stochastic terms that conserve kinetic
energy of each particle. In a large-time weak-coupling limit, the energies
of the particles evolves autonomously following a (non-gradient) stochastic
Ginzburg-Landau dynamics. Then a non linear heat equation can be deduced
from this stochastic dynamics under a hydrodynamic diffusive limit. This is
a joint work with Carlangelo Liverani.
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October 16
Midwest Probability Colloqium Northwestern University
No seminar
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October 23
Elena Kosygina Baruch College, CUNY
Limit laws of excited random walks on integers
We consider excited random walks on integers with a bounded number of
i.i.d. cookies per site without the non-negativity assumption on the
drifts induced by the ``cookies''. We shall discuss known and new
results about limit laws of these random walks (under the averaged
measure) as well as some open questions. (arXiv:0908.4356v1 [math.PR])
(joint with T. Mountford, EPFL, Lausanne)
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October 30
Larry Gray UMN
From Weak to Strong: putting the 'S' onto the LLN
The main result of the talk is a Strong Law of Large Numbers for
a class of particle systems for which the LLN comes somewhat naturally from
coupling arguments. But the talk will start with much simpler examples of
how one can get from the LLN to the SLLN, including a version of
coin-tossing that could be called the "Diaconis Coin".
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November 6
Sunder Sethuraman Iowa State
Some asymptotics in preferential attachment random graphs
Consider the following growth rule: Start with an initial connected
graph G_0. At time n>0, a new vertex is attached to one of the vertices in G_
{n-1} with probability proportional to its degree, and the new formed graph is
labeled G_n. Such processes and variants are known as `preferential
attachment' schemes.
One of the main results is a LLN for the number of vertices in {G_n} with a
fixed degree (Bollobas et al 2001). In this talk, we will discuss an embedding
into branching processes of the preferential attachment scheme from which the
LLN can also be recovered. We will also discuss large deviations with respect
to the number of `leaves' (vertices with degree 1).
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November 13
Hyejin Kim IMA
Weak convergence of one dimensional Markov
processes governed by the
W. Feller generalized second order differential operator in closed
intervals.
It is well known that solutions of classical initial--boundary
problems for second order parabolic equations depend continuously
on the coefficients if the coefficients converge to their limits
in a strong enough topology.
In case of one spatial variable, we consider the question of
the weakest possible topology providing convergence of the
solutions. The problem is closely related to weak convergence of
corresponding diffusion processes. Continuous Markov processes
corresponding to the Feller operators $D_vD_u$ arise, in general,
as limiting processes. Solutions of the parabolic equations
converge, in general, to the solutions of corresponding
initial--boundary problems for the limiting operator $D_vD_u$.
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November 20
Tiefeng Jiang UMN
Spectral properties of large random graphs
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November 27
No seminar this week.
Happy Thanksgiving
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December 4
Hye-Won Kang UMN
The optimal size for space discretization in spatially nonuniform
reaction-diffusion systems
In this talk, I will discuss how to discretize space to model
stochastic reaction-diffusion systems. A system with chemical reactions and
diffusion is modeled using a continuous time Markov jump process. Diffusion
is described as a jump to the neighboring compartments with proper spatial
discretization. Considering stationary mean and variance of each species in
each compartment, the optimal size for spatial discretization will be
suggested. Then, I will show criteria to discretize the corresponding
deterministic reaction-diffusion equation for concentration of species. The
optimal size for spatial discretization obtained from the deterministic case
coincides with the result of the stochastic case. This is a joint work with
Hans Othmer and Likun Zheng.
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December 11
Louis-Pierre Arguin Courant Institute
TBA
Spring 2010
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January 22
TBA TBA
TBA
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January 29
TBA TBA
TBA
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February 5
TBA TBA
TBA
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February 12
TBA TBA
TBA
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February 19
Jason Miller Stanford University
TBA
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February 26
Claude Lebris ENPC
TBA
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March 5
Jie Xiong University
of Tennessee
TBA
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March 12
TBA TBA
TBA
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March 19
No seminar this week.
Spring Break
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March 26
Erwin Bolthausen
University of Zurich
TBA
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April 2
TBA TBA
TBA
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April 9
Steve Zelditch John Hopkins and Northwestern
TBA
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April 16
TBA TBA
TBA
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April 23
TBA TBA
TBA
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April 30
TBA TBA
TBA
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May 7
TBA TBA
TBA
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