10th Probability Day Erlangen-München

• Organizers: Nina Gantert (München), Hans-Otto Georgii (München), Andreas Greven (Erlangen), Gerhard Keller (Erlangen), Franz Merkl (München), Silke Rolles (München), and Vitali Wachtel (München).
• The probability day takes place on Friday, July 1, 2011 in room 349 at the mathematical institute of the Ludwig-Maximilian-Universität München. Information how to get to the mathematical institute can be found here. It takes approximately 25 minutes to walk from the train station to the mathematical institute.

Program

• 14:00 - 15:00 Rob van den Berg (Amsterdam): Extensions of the BK inequality
  Abstract: The BK inequality, proved by van den Berg and Kesten in 1984 says that, for product measures on {0,1}^n, the probability that two increasing events `occur disjointly' is smaller than or equal to the product of the two individual probabilities. This result is often used in percolation and interacting particle systems. Their conjecture that the inequality even holds for all events was proved by Reimer in 1994.
  In spite of Reimer's work, several natural, fundamental problems in this area remained open. During this talk I will discuss some very recent progress, in particular an extension of the BK inequality to randomly drawn subsets of fixed size (joint work with Johan Jonasson). I will also mention a modified version of the notion `disjoint occurrence' for the Ising model (work in progress with Alberto Gandolfi).
• 15:00 - 16:00 Jiri Cerny (Zürich): Vacant set of random walk on (random) graphs.
  Abstract: The vacant set is the set of vertices not visited by a random walk on a graph before a given time T. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit' when |G| and T grow simultaneously), and the relation of the problem to the random interlacement percolation. I will then concentrate on the case when G is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.
• 16:00 - 16:30 Coffee and tea
• 16:30 - 17:30 Frank Aurzada (Berlin): The one-sided exit problem for integrated processes and fractional Brownian motion
  Abstract: We study the one-sided exit problem problem, also known as one-sided barrier problem, that is, given a stochastic process X we would like to find, as T\to\infty, the asymptotic rate of P( \sup_{0\leq t\leq T} X_t \leq 1 ). This question is considered for alpha-fractionally integrated centered L\'evy processes and 'integrated' centered random walks. We show that the rate of decrease of the above probability is polynomial with exponent theta=theta(alpha)>0 which only depends on alpha but not on the choice of the L\'evy process or random walk.
  This generalizes results of Y.G. Sinai (1991) who considered alpha=1 and the simple random walk. Similar recent results are due to V. Vysotsky (2010) and A. Dembo and F. Gao (2011).
  Finally, the results are compared to the corresponding ones for fractional Brownian motion.
• After the talks, there will be a dinner.