abstract:
In this lecture
we will discuss random walks in random environment. We will present some of
the progresses that have been made in the case of ballistic walks, i.e. when
the walks have a non degenerate limiting velocity. The so-called conditions
(T) and (T'), have been helpful in this situation, and may turn out to
characterize the ballistic phase of the model when the space dimension is 2
or more.
abstract:
The introduction of the Schramm-Loewner Evolution (SLE) has led to the identification of the scaling limit of two-dimensional critical percolation. For critical site percolation on the triangular lattice, this identification can be made rigorous both for a single interface and for the full scaling limit (i.e., the collection of all cluster boundaries).
The full scaling limit is described by a loop process in the plane introduced by C. M. Newman and myself. After a brief discussion of some properties of this loop process, I will take it as the starting point to present some mostly heuristic and conjectural new developments (and some open questions) concerning the near-critical scaling limit of 2D percolation and related models (such as the minimal spanning tree).
If time permits, I will discuss certain connections with the renormalization group, and a type of conformal covariance that replaces the full conformal invariance typical of systems at the critical point.
(Joint work with L. R. Fontes and C. M. Newman.)