A shape theorem for Riemannian first-passage percolation

Riemannian first-passage percolation is a continuum model, with a distance function arising from a random Riemannian metric in ^Rd. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability of 1.