13th Probability Day Erlangen-München
- Organizers: Noam Berger (München), 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, May 23, 2014, in room B 005 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 Aernout van Enter ^{} (Groningen): Bootstrap percolation, the role of anisotropy: Questions, some answers and applications.
Abstract: Bootstrap percolation models are Cellular Automata, with deterministic growth rules, but random initial conditions. They describe growth processes, in which in a metastable situation nucleation occurs from the creation of some kind of critical droplet. Such droplets are rare, but once they appear, they grow to cover the whole of space.
The occurrence of such critical droplets in large volumes is ruled by asymptotic probabilities. We describe some known results on the asymptotics for such models. Moreover, we discuss how the scaling of these probabilities with the volume is modified in the presence of anisotropy. We also discuss why numerics have a rather bad track record in the subject.
This is based on joint work with Tim Hulshof, Hugo Duminil-Copin, Rob Morris and Anne Fey. - 15:00 - 16:00 Peter Imkeller ^{} (Berlin): A Fourier analytic approach of integration Abstract
- 16:00 - 16:30 Coffee and tea
- 16:30 - 17:30 Antal Jarai ^{} (Bath): Electrical resistance of the low dimensional critical branching random walk
Abstract: In his seminal work Kesten (1986) initiated the rigorous study of random walk on random fractals, and showed that the simple random walk on critical 2D percolation clusters is sub-diffusive, i.e. the time it takes for the walk to exit a ball of radius R is of larger order than R^2. He also proved a similar result for random walk on a critical branching process family tree, where the time it takes to exit a ball of radius R was found to be of order R^3. Several other critical exponents for the latter case (that can be regarded as "mean-field") were proved by Barlow and Kumagai (2006), and later extended to critical oriented percolation clusters in dimensions d > 6 by Barlow, Jarai, Kumagai and Slade (2008).
Here we consider the case d < 6 in the similar, but simpler, setting of a critical branching random walk cluster, and rigorously prove that the behaviour is different than in the mean-field case d > 6. In particular, the time it takes for the walk to exit a ball of radius R is at most R^{3-alpha} for some alpha > 0. The result is achieved by showing that the electrical resistance between the origin and generation R in the branching random walk cluster is at most O(R^{1-alpha}) for some alpha > 0. Our lower bound on the exponent alpha is universal with respect to the offspring distribution and the walk step-distribution, under some moment assumptions.
(Joint work with Asaf Nachmias). - We will go for dinner after the talks.