# A Sharp Tail Bound for the Expander Random Sampler

## TL;DR

This paper introduces a new, sharper tail bound for the number of marked vertices encountered in a random walk on an expander graph, improving previous concentration results especially for moderate marking fractions.

## Contribution

It provides a novel and more precise tail bound for the distribution of marked vertices in expander graph random walks, enhancing understanding of their probabilistic behavior.

## Key findings

- The new tail bound is sharper than previous bounds for certain marking fractions.
- The result improves concentration estimates for random walks on expanders.
- Applicable to analyzing randomized algorithms and network processes on expanders.

## Abstract

Consider an expander graph in which a $\mu$ fraction of the vertices are marked. A random walk starts at a uniform vertex and at each step continues to a random neighbor. Gillman showed in 1993 that the number of marked vertices seen in a random walk of length $n$ is concentrated around its expectation, $\Phi := \mu n$, independent of the size of the graph. Here we provide a new and sharp tail bound, improving on the existing bounds whenever $\mu$ is not too large.