  In this paper we prove the first quantitative convergence rates for the graph
infinity Laplace equation for length scales at the connectivity threshold. In
the graph-based semi-supervised learning community this equation is also known
as Lipschitz learning. The graph infinity Laplace equation is characterized by
the metric on the underlying space, and convergence rates follow from
convergence rates for graph distances. At the connectivity threshold, this
problem is related to Euclidean first passage percolation, which is concerned
with the Euclidean distance function $d_{h}(x,y)$ on a homogeneous Poisson
point process on $\mathbb{R}^d$, where admissible paths have step size at most
$h>0$. Using a suitable regularization of the distance function and
subadditivity we prove that ${d_{h_s}(0,se_1)}/ s \to \sigma$ as $s\to\infty$
almost surely where $\sigma \geq 1$ is a dimensional constant and $h_s\gtrsim
\log(s)^\frac{1}{d}$. A convergence rate is not available due to a lack of
approximate superadditivity when $h_s\to \infty$. Instead, we prove convergence
rates for the ratio $\frac{d_{h}(0,se_1)}{d_{h}(0,2se_1)}\to \frac{1}{2}$ when
$h$ is frozen and does not depend on $s$. Combining this with the techniques
that we developed in (Bungert, Calder, Roith, IMA Journal of Numerical
Analysis, 2022), we show that this notion of ratio convergence is sufficient to
establish uniform convergence rates for solutions of the graph infinity Laplace
equation at percolation length scales.
