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Probability Seminar List
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The CUNY Probability Seminar is typically held on Tuesdays at 4pm in the CUNY Graduate Math Department. The exact dates, times and locations are mentioned below.

Seminar List for Fall 2008

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Sep 16, 2008 4:00pm, Room 5417Edit Delete B/W(color)Hard CopyEmail Entry
Speaker Jay Rosen, CUNY/ College of Staten Island
Title A CLT for the L^2 modulus of continuity of local times.
Sep 23, 2008 4:00pm, Room 5417Edit Delete B/W(color)Hard CopyEmail Entry
Speaker Jingchen Liu, Columbia University
Title Rare-event Simulation for Heavy-tailed Multi-server Queue
Abstract
In this talk, I will present the first provably efficient simulation
algorithm, via state-dependent importance sampling, to compute the
probability that a customer experiences a long delay for a positive
recurrent two-server (G/G/2) queue with heavy-tailed service
requirement.
Such a delay is usually caused by one or two customers (depending on
the
traffic intensity) who have extremely large service requirement and
occupy
the servers for a long time. We propose a three-step program to design
the
algorithm and prove its efficiency. First, we adopt a mixture family of

changes-of-measure; second, propose an appropriate Lyapunov inequality
to
control the variance of our estimator; third, construct a Lyapunov
function
(the solution to the Lyapunov inequality) and tune various parameters
to
verify the inequality. Because of the upper bound provided by the
Lyapunov
function, our method also suggests an asymptotic approximation of the
rare-event probability. Therefore, rare-event simulation and large
deviations analysis are connected naturally. Our strategy including the

mixture family, the construction of Lypunov function, and proof
techniques
can solve a large class of problems
Oct 7, 2008 4:00pm, Room 5417Edit Delete B/W(color)Hard CopyEmail Entry
Speaker Emanuel Milman, Institute for Advanced Study, Princeton
Title A stability result for the mixing time of Brownian motion with boundary reflectance on a convex domain.
Abstract It is classical that the L2 mixing time of Brownian motion on a nice domain in Euclidean space with reflectance on its boundary, is inversely proportional to the spectral gap of the Neumann Laplacian on that domain.
For convex domains, we provide a new characterization of the spectral gap, by showing that Cheeger's isoperimetric inequality, the spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and any arbitrarily slow (but fixed) tail-decay of these functions, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, Gromov-Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap on convex domains under convex perturbations which preserve volume (up to constants) and under maps which are "on-average" Lipschitz. This also leads to the following characterization of the square root of the L2 mixing time of Brownian motion on convex domains: it is equivalent (up to constants) to the expectation (with respect to the uniform measure on the domain) of the distance from the "worst" Borel set having measure 1/2. In addition, we easily recover (and extend) many previously known lower bounds on the spectral gap of convex domains, due to Payne-Weinberger, Li-Yau, Kannan-Lovász-Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry-Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,¥) curvature-dimension condition of Bakry-Émery.
Oct 28, 2008 4:00pm, Room 5417Edit Delete B/W(color)Hard CopyEmail Entry
Speaker Ross Pinsky, Technion
Title Spectral Calculations for Diffusion operators
Dec 2, 2008 4:00pm, Room 84??Edit Delete B/W(color)Hard CopyEmail Entry
Speaker Pawel Hitczenko, Drexel University
Title Tails of perpetuities
Abstract By perpetuity we mean a random variable R satisfying the equation
R d
=
 
 Q+MR,
where (Q,M) are random variables independent of R. Alternatively, R is a limit, in distribution, of a sequence (Rn) satisfying
Rn=Qn+MnRn-1,    n ³ 1,
(\theequation)
where (Qn,Mn) are iid copies of (Q,M), (Qn,Mn) is independent of Rn-1, and R0 is arbitrary. Accordingly, R may be written as
R d
=
 
 ¥
å
i=1 
 Qi i-1
Õ
j=1 
 Mi.
(\theequation)
Conditions guaranteeing convergence in (1) and (2) have been given by Kesten. Equations like (1) are ubiquitous in applied mathematics.
The main focus of research has been on the tail behavior of R:
P(|R| ³ x),    as    x®¥.
The case P(|M| > 1) > 0 was analysed by Kesten who showed that R is always heavy-tailed. The complementary case 0 £ | M| £ 1 is much less understood. Goldie and Grübel showed that in that case, the tails are never heavier than exponential and that if |M| behaves near 1 as a uniform random variable then the tails of R are Poissonian.
In this talk we will present further results about the tails of R and their connection to the behavior of |M| near 1.
This is a based on a joint work with Jacek Wesoowski, Technical University of Warsaw.
Dec 9, 2008 4:00pm, Room 84??Edit Delete B/W(color)Hard CopyEmail Entry
Speaker Julien Dubedat,
Title Dimers and analytic torsion
Abstract
We discuss Gaussian invariance principles for dimer models in relation
with variational formulae for zeta-determinants of Cauchy-Riemann
operators.

an introductory text: http://arxiv.org/pdf/math/0310326v1