# On Covering Monotonic Paths with Simple Random Walk

## TL;DR

This paper investigates the probability that a simple random walk covers a monotonic path in high dimensions, identifying the path that maximizes this probability and providing bounds on its decay rate.

## Contribution

It establishes that the monotonic increasing path maximizes covering probability and derives exponential bounds for the decay rate in dimensions four and higher.

## Key findings

- Monotonic increasing paths maximize covering probability.
- Exponential upper bounds on decay rate for dimensions ≥ 4.
- Covering probability decreases exponentially with path length.

## Abstract

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\ge 4$.