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Abstract:  Approximating a high-dimensional lattice by a regular tree ("Bethe lattice")  is a classical device in Math Physics. But how do we best approximate the local geometry of a random cloud of points in space?   The answer is the Poisson-weighted infinite tree (PWIT),  introduced by Aldous and Steel for a completely different purpose. 

We exhibit the effectiveness of this approximation by analyzing the greedy matching of two independent Poisson processes in Euclidean space under an asymmetric color restriction.  Blue points can only match to red points, while red points can match to points of either color.  It is unknown if there exist intensities for the red and blue processes for which all points are matched.  We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension.  Our proof uses greedy matching on the PWIT, which can be analyzed via differential equations. The proof is inspired by Aldous' 2002 proof of the zeta(2) limit for the random assignment problem, which was motivated by the heuristic cavity argument of Mezard-Parisi (1987). 

(Joint work with  Alexander Holroyd and James  Martin.)