Projects per year
Abstract
Consider a randomlyoriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic walk is then started at the origin and at each step moves diagonally to the nearest vertex in the direction of the horizontal and vertical lines of the present location. 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, almost surely. 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 subexponentially in n. It is easy to show that higher dimensional paths may not localize on two vertices but will still eventually become periodic, and are therefore bounded.
Original language  English 

Article number  137 
Number of pages  20 
Journal  Electronic Journal of Probability 
Volume  24 
DOIs  
Publication status  Published  1 Jan 2019 
Keywords
 82C41
 AMS MSC 2010: 60K37
 Dependent percolation
 Deterministic walk in random environment
Projects
 1 Finished

Random walks with long memory
Collevecchio, A., Garoni, T., Hamza, K., Cotar, C. & Tarres, P.
Australian Research Council (ARC)
1/05/18 → 1/05/22
Project: Research