Consider a point process in Euclidean space obtained by perturbing the integer lattice with IID Gaussians. Our goal is to match the perturbed points with the points of the lattice in an invariant way and such that the typical matching distance is as small as possible (or nearly so). One obvious attempt to do this is to send each perturbed point back to its original location. This gives an invariant matching whose matching distance has Gaussian tails. Is this best possible? It turns out that it is not -- there are better ways of doing this. We establish the optimal tail bound and construct an invariant matching that achieves it. Time permitting, we will discuss an analogous question in the hyperbolic plane and/or regular trees.
Based on joint works with Dor Elboim and Oren Yakir.