# A deterministic walk on the randomly oriented Manhattan lattice

## TL;DR

This paper studies a deterministic walk on a randomly oriented Manhattan lattice, showing that in two dimensions it localizes on two vertices almost surely and provides tail estimates for path lengths, with higher dimensions leading to periodic paths.

## Contribution

It introduces a novel deterministic walk model on a randomly oriented lattice and proves localization on two vertices in two dimensions, with tail estimates for path lengths.

## Key findings

- In two dimensions, the walk localizes on two vertices almost surely.
- The probability of the path length exceeding n decays sub-exponentially.
- Higher dimensions lead to periodic, bounded paths instead of localization.

## Abstract

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.