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] and directed last-passage percolation.^[4]

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

↑ Busemann, Herbert. The geometry of geodesics. Vol. 6. DoverPublications. com, 1985.
↑ Hoffman, Christopher. "Coexistence for Richardson type competing spatial growth models." The Annals of Applied Probability 15.1B (2005): 739-747.
↑ Damron, Michael, and Jack Hanson. "Busemann functions and infinite geodesics in two-dimensional first-passage percolation." arXiv preprint arXiv:1209.3036 (2012)
↑ Nicos Georgiou, Firas Rassoul-Agha, and Timo Seppäläinen. "Geodesics and the competition interface for the corner growth model." arXiv preprint arXiv:1510.00860 (2015)

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