# Directed random walks on polytopes with few facets

## TL;DR

This paper proves that directed random walks on simple polytopes with few facets, specifically grid graphs, terminate within a polylogarithmic number of steps under various orientations, extending previous linear edge orientation results.

## Contribution

It generalizes the termination bound of directed random walks from linear to unique sink orientations on polytopes with two facets difference.

## Key findings

- Directed random walks terminate after O(log^2 n) steps.
- Bound holds for both linear and unique sink orientations.
- Extends previous results to more general orientations.

## Abstract

Let $P$ be a simple polytope with $n-d = 2$, where $d$ is the dimension and $n$ is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of $P$ terminates after $O(\log^2 n)$ steps, if the edges are oriented in a (pseudo-)linear fashion. We prove that the same bound holds for the more general unique sink orientations.