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

Understand

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.