Busemann function - Alchetron, The Free Social Encyclopedia

Busemann functions were introduced by Herbert 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 γ : [ 0 , ∞ ) → X which minimizes distance everywhere along its length. i.e., for all t , t ′ ∈ [ 0 , ∞ ) ,

d ( γ ( t ) , γ ( t ′ ) ) = | t − t ′ | .

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

Given a ray γ, the Busemann function B γ : X → R is defined by

B γ ( x ) = lim t → ∞ ( d ( γ ( t ) , x ) − t )

That is, when t is very large, the distance d ( γ ( t ) , x ) is approximately equal to B γ ( 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 function Wikipedia

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