Annealed random walk conditioned on survival among Bernoulli obstacles

Abstract: I will present two recent results on a discrete time random walk conditioned to avoid Bernoulli obstacles on the d-dimensional integer lattice obtained in joint works with Jian Ding, Rongfeng Sun and Changji Xu. The first result on this model dates back to a famous work by Donsker and Varadhan on the Wiener sausage in 1975. Since then, it has been intensively studied and various localization results have been proved. In particular, the random walk is known to localize in a ball of sub-diffusive size under the annealed law. Our first result gives a more detailed geometric description of the range of the random walk. More precisely, we showed that it completely fills the ball where the walk is localized, and in addition we got a sharp estimate on the size of its boundary.

Our second result is about the response to an external force. If we give a bias to the random walk, then the model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. This phase transition was proved by Sznitman and later, Ioffe and Velenik studied the ballistic phase in detail. In the sub-ballistic phase, physicists conjectured that the walk is localized in a sub-diffusive scale as in the unbiased case, but it has not been proved. We prove this conjecture with a precise information on the behavior of whole path.

probability

Audience: advanced learners

Bangalore Probability Seminar

Series comments: The link to zoom meeting can be found on the seminar's google calendar - www.isibang.ac.in/~d.yogesh/BPS.html