Busemann function

Busemann functions were introduced by Busemann to study the large-scale geometry of metric spaces in his seminal The Geometry of Geodesics. More recently, Busemann functions have been used by probabilists to study asymptotic properties in models of first-passage percolation and directed last-passage percolation.

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.