First passage percolation on $\mathbb{Z}^2$ -- a simulation study

TL;DR

This simulation study investigates first passage percolation on a 2D lattice, revealing how passage time distributions influence growth speed and shape, with key metrics identified through large-scale computational experiments.

Contribution

The paper provides new insights into the relationship between passage time distributions and growth dynamics in first passage percolation, emphasizing the role of the expected minimum passage time.

Findings

The inverse growth speed (time constant) is linearly related to the expected minimum of passage times.

Directional time constants increase from axes to diagonal, shaping the infected region.

The infected shape approaches a circle as distribution variability increases.

Abstract

First passage percolation on $\mathbb{Z}^2$ is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges. In this paper, the speed of the growth and the shape of the infected set is studied by aid of large-scale computer simulations, with focus on continuous passage time distributions. It is found that the most important quantity for determining the value of the time constant, which indicates the inverse asymptotic speed of the growth, is $\mathbf{E}[\min\{\tau_1,\ldots,\tau_4\}]$, where $\tau_1,\ldots,\tau_4$ are i.i.d. passage time variables. The relation is linear for a large class of passage time distributions. Furthermore, the directional time constants are seen to be increasing when moving from the axis towards the diagonal, so that the limiting shape is contained in a circle with radius defined by the speed along the axes. The shape comes closer to the circle for distributions with larger variability.