Busemann function

Busemann functions were introduced by Busemann to study the large-scale geometry of metric spaces in his seminal The Geometry of Geodesics.^[1] More recently, Busemann functions have been used by probabilists to study asymptotic properties in models of first-passage percolation.^[2][3]

Definition

Let $(X,d)$ be a metric space. A ray is a path $\gamma : [0,\infty) \to X$ which minimizes distance everywhere along its length. i.e., for all $t,t' \in [0,\infty)$,

$d\big(\gamma(t), \gamma(t') \big) = \big| t - t' \big|$.

Equivalently, a ray is an isometry from the "canonical ray" (the set $[0,\infty)$ equipped with the Euclidean metric) into the metric space X.

Given a ray γ, the Busemann function $B_\gamma : X \to \mathbb R$ is defined by

$B_\gamma(x)=\lim_{t\to\infty}\big( d\big( \gamma(t), x \big) - t \big)$

That is, when t is very large, the distance $d\big( \gamma(t), x \big)$ is approximately equal to $B_\gamma(x) + t$. Given a ray γ, its Busemann function is always well-defined.

Loosely speaking, a Busemann function can be thought of as a "distance to infinity" along the ray γ.

References

1. Busemann, Herbert. The geometry of geodesics. Vol. 6. DoverPublications. com, 1985.
2. Hoffman, Christopher. "Coexistence for Richardson type competing spatial growth models." The Annals of Applied Probability 15.1B (2005): 739-747.
3. Damron, Michael, and Jack Hanson. "Busemann functions and infinite geodesics in two-dimensional first-passage percolation." arXiv preprint arXiv:1209.3036 (2012).