2020/01/13 by Daniel Ahlberg, Ahlberg, Daniel
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2001.04142
openalex publication_date 2020/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.