OPTIMAL CHEEGER CUTS AND BISECTIONS OF RANDOM GEOMETRIC GRAPHS

Let d >= 2. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on n random points in a d-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of n) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large n to an analogous Cheeger-type constant of the domain. Previously, Garcia Trillos et al. had shown this for d >= 3 but had required an extra condition on the distance parameter when d = 2.