First-passage percolation on random simple triangulations

TL;DR

This paper investigates how first-passage percolation distances behave on random simple triangulations and their dual maps, showing that these distances are tightly concentrated around a constant multiple of the graph distance.

Contribution

It demonstrates that first-passage percolation distances on random triangulations concentrate in a small window around a scaled graph distance, providing new insights into their geometric properties.

Findings

First-passage percolation distances concentrate in an $o_p(n^{1/4})$ window.

Distances are tightly concentrated around a constant multiple of the graph distance.

Results apply to both triangulations and their dual maps.

Abstract

We study first-passage percolation on random simple triangulations and their dual maps with independent identically distributed link weights. Our main result shows that the first-passage percolation distance concentrates in an $o_p(n^{1/4})$ window around a constant multiple of the graph distance.