Point Configurations in Brownian Traces

Abstract

We study finite and countable point configurations in the trace of Brownian motion in $\mathbb{R}^d$, $d\geq2$. In the plane, the set of translations placing a prescribed finite configuration on the trace has Hausdorff dimension two in every nonempty open set; we prove analogous dimension-preservation results for radial and bilipschitz patterns, and every prescribed countable configuration has a dense translation set. For $d\geq3$ we prove a sharp occurrence criterion for regular scale-invariant families of configurations, in terms of the ranks of all labelled subconfigurations. It gives exact thresholds for arithmetic progressions and for homothetic and similar copies of finite sets. In particular we recover and extend a result of Benjamini and Kozma: almost surely the longest arithmetic progression in the trace has length five in $\mathbb{R}^3$, three in $\mathbb{R}^4$ and $\mathbb{R}^5$, and two in $\mathbb{R}^d$ for $d\geq6$.