First-passage percolation and local modifications of distances in random triangulations

Understand

We study local modifications of the graph distance in large random triangulations.

• Our main results show that, in large scales, the modified distance behaves like a deterministic constant $\mathbf{c}~\in~(0,\infty)$ times the usual graph distance.
• This applies in particular to the first-passage percolation distance obtained by assigning independent random weights to the edges of the graph.
• We also consider the graph distance on the dual map, and the first-passage percolation on the dual map with exponential edge weights, which is closely related to the so-called Eden model.