A Gaussian fixed point random walk

TL;DR

This paper introduces a discrete random walk on the real line with specific step distributions that converges to a standard normal distribution, leading to new online discrepancy bounds and efficient algorithms for the Komlós conjecture.

Contribution

It designs a novel random walk with particular step probabilities that achieves a Gaussian stationary distribution, enabling new discrepancy bounds and algorithms.

Findings

Achieves a Gaussian stationary distribution with steps mostly in \\pm 1.

Provides an online discrepancy result for partial colorings with \\pm 1, 2 signs.

Recovers linear time algorithms for the Komlós conjecture in an online setting.

Abstract

In this note, we design a discrete random walk on the real line which takes steps $0, \pm 1$ (and one with steps in $\{\pm 1, 2\}$) where at least $96\%$ of the signs are $\pm 1$ in expectation, and which has $\mathcal{N}(0,1)$ as a stationary distribution. As an immediate corollary, we obtain an online version of Banaszczyk's discrepancy result for partial colorings and $\pm 1, 2$ signings. Additionally, we recover linear time algorithms for logarithmic bounds for the Koml\'{o}s conjecture in an oblivious online setting.