Existence and Coexistence in First-Passage Percolation

Abstract

We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in first-passage percolation and study their implications for the multi-type Richardson model. In two dimensions this establishes a dual relation between the existence of infinite geodesics and coexistence among competing types. The argument amounts to making precise the heuristic that infinite geodesics can be thought of as ‘highways to infinity’. We explain the limitations of the current techniques by presenting a partial result in dimensions d > 2.