TY - JOUR

T1 - A deterministic walk on the randomly oriented Manhattan lattice

AU - Collevecchio, Andrea

AU - Hamza, Kais

AU - Tournier, Laurent

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Consider a randomly-oriented 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 sub-exponentially 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.

AB - Consider a randomly-oriented 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 sub-exponentially 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.

KW - 82C41

KW - AMS MSC 2010: 60K37

KW - Dependent percolation

KW - Deterministic walk in random environment

UR - http://www.scopus.com/inward/record.url?scp=85076274059&partnerID=8YFLogxK

U2 - 10.1214/19-EJP385

DO - 10.1214/19-EJP385

M3 - Article

AN - SCOPUS:85076274059

VL - 24

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 137

ER -