Tertiles and the time constant

TL;DR

This paper investigates bounds on the time constant in planar first-passage percolation using tertiles of the weight distribution, providing a counter-example to a previously posed problem.

Contribution

It introduces bounds on the time constant based on tertiles and presents a counter-example to an existing open problem in the field.

Findings

The time constant can be bounded by multiples of the first and second tertiles.

A counter-example to a problem by Alm and Deijfen is constructed.

The approach links distribution tertiles to percolation time constants.

Abstract

We consider planar first-passage percolation and show that the time constant can be bounded by multiples of the first and second tertiles of the weight distribution. As a consequence we obtain a counter-example to a problem proposed by Alm and Deijfen.