Existence and coexistence in first-passage percolation (2001.04142v2)

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 higher than two.