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

TL;DR

This paper investigates how local modifications of graph distances in large random triangulations, including first-passage percolation, behave like scaled versions of the usual distance, with explicit constants computed in certain models.

Contribution

It establishes that modified distances in large random triangulations are asymptotically proportional to the usual graph distance and computes the proportionality constant in specific cases.

Findings

Modified distances behave like a constant times the usual distance at large scales.

Explicit constants are computed for first-passage percolation and dual map distances.

Balls in the UIPT under modified distances are asymptotically close to those under the standard graph distance.

Abstract

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. In the latter two cases, we are able to compute explicitly the constant $\mathbf{c}$ by using earlier results about asymptotics for the peeling process. In general however, the constant $\mathbf{c}$ is obtained from a subadditivity argument in the infinite half-plane model that describes the asymptotic shape of the triangulation near the boundary of a large ball. Our results apply in particular to the infinite random triangulation known as the UIPT, and show that balls of the UIPT for the modified distance are asymptotically close to balls for the graph distance.