Luke Turvey

Minimal matchings for Poisson point processes

Suppose that red and blue points form independent homogeneous Poisson processes of unit intensity on R^d. For a positive (respectively, negative) parameter \gamma we consider red-blue matchings that locally minimise (respectively, maximise) the sum of the \gamma^th powers of edge lengths, subject to locally minimising the number of unmatched points. These matchings were first introduced and studied by Holroyd, Janson and Wastlund, where the \gamma-minimal matchings in dimension one were almost completely categorised for all \gamma \in [\infty, \infty]. A complete classification in dimensions d > 1 is still unknown. We will present a similiar classification to dimension 1 for  \gamma-minimal matchings on the strip R x [0,1] where the picture is almost completely the same. We further provide a tight tail bound on the typical matching distance when \gamma>1 in the one-colour case.