Consider a randomly-oriented two dimensional Manhattan lattice, where to each horizontal and vertical ‘line’, is assigned, once and for all, a random direction by flipping i.i.d. coins. Consider the following random process. It starts from the origin and at each step it moves diagonally to one of the vertices at distance √2 from the present one. The unique choice simply follows the orientation of the lines incident to the present location of the process. This definition can be generalized, in a natural way, to larger dimensions, but we mainly focus on the two dimensional case. In this context the process localizes on two vertices at all large times, a.s. We also provide estimates for the tail of the length of paths, when the walk is defined on the two dimensional lattice. In particular, the probability of the path to be larger than n decays sub-exponentially in n. It is easy to show that localization happens also at larger dimensions, but not necessarily on two vertices. Joint work with K. Hamza and L. Tournier.
Professor Collevecchio is a Senior Lecturer at Monash University, Australia. He received his Ph.D. from the Purdue University, after which he has been a Post-doc at the University of Chieti and at the Max Planck Institute in Leipzig, and Assistant Professor at the University of Venice. His research areas include interacting processes, large deviations, random graphs, and study of mixing times for Markov chains.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai