--Probability Seminar is on Fridays at 3:30 PM in VinH 206.
Feb 11: Nicolai Krylov
Feb 18: Yevgeniy Kovchegov
Feb 25: Alexander Tikhomirov
March 4: Anastasia Ruzmaikina
March 11: Ofer Zeitouni
March 18: No seminar, semester break
March 25: Naresh Jain
April 1: Stas Volkov
April 8: No seminar, Riviere-Fabes Symposium
April 15: Alexander Tikhomirov
April 22: Open
April 29: Professor Hu
May 6: Open, last day of instruction.
--Organizer: Naresh Jain
Speaker: Anastasia Ruzmaikina
Title: "Characterization of the invariant measures at the leading edge for the competing particle systems."
Speaker: Ofer Zeitouni
Title: "Spanning trees and random walks in random environments."
Abstract: We define a notion of ancestral functions for spanning trees, and
prove an a-priori estimate on the length of ancestral lines in random
stationary spanning tress in Z^d. This is then used to construct a counter
example for a conjectured 0-1 law for random walks in uniformly eliptic mixing
random environments. The 0-1 conjecture remains open for the most interesting
i.i.d. setup. (joint work with Maury Bramson and Martin Zerner)
Speaker: Naresh Jain
Title: "Large deviations for occupation time measures of Markov
processes with $L_2$ semigroups"
Summary: We consider a continuous-time Markov process with a
nonmetrizable state space. The process is assumed to have a
$\sigma$-finite invariant measure $\lambda$ and the associated semigroup is
assumed to be strongly continuous in $L_2(\lambda)$. Large deviation upper
and lower bounds are obtained with a certain rate function. Under the
additional assumption of self-adjointness of the semigroup, we obtain a
result which subsumes many known results. Corresponding results for
discrete time are also obtained. (This is joint work with Nikolai Krylov)
Speaker: Stas Volkov
Title:"5x+1 : a probabilistic view"
Abstract: A notorious 3x+1 conjecture, also know as Collatz Problem, states
that the following algorithm always converges to 1: take any positive integer,
if it is even, divide it by two or if it is odd multiply it by three and add
one. Then keep repeating the algorithm infinitely, unless you get one. A
path is thus something like 13 - 40 - 20 - 10 - 5 -16 -8 - 4 - 2 - 1.(Hint:
try to work this out with 27.) Heuristically it is fairly obvious that any
number will end up at 1, since on average each even number is divisible by the
2nd power of 2, and hence in three consecutive steps "on average" x reduces to
approximately 3/4 x.
At the same time, if we multiplied x by 5 and not by 3, then "on average" we
should drift off to infinity. Numerically, however, it is suggested that some
small number Q(N) of integers between 1 and N (large) still goes to one of
three cycles and thus does not diverge under 5x+1 algorithm.
We construct a probabilistic model, related to a random environment on trees
and first passage percolation, which "imitates" the algorithm and gives the
asymptotic behavior of the analogue of Q(N). At the same time, unfortunately,
only probabilistic statements can be proved rigorously.
Some interesting relevant information can be found at
http://www.ieeta.pt/~tos/px+1.html and
http://mathworld.wolfram.com/CollatzProblem.html
Speaker: Alexander Tikhomirov
Title:"On the rate of convergence in some limit theorem for spectra of random
matrices"
We consider two classes of random matrices:
1)class of Wigner matrices -- real symmetric or Hermitian matrices with
independent (except symmetry or Hermitian) entries ;
2)class of sample covariance matrices introduced by Wishart.
We discuss the bounds for the Kolmogorov distance between the expected
spectral distribution function of random matrix and corresponding limit
distribution function. We consider also the bounds for Kolmogorov distance
between empirical spectral distribution function and corresponding limit
spectral distribution function.
The discussing results are joint results with Professor F. Goetze
(Bielefeld, Germany)