---
abstract: 'An Euclidean first-passage percolation (FPP) model describing the competing growth between $k$ different types of infection is considered. We focus on the long time behavior of this multi-type growth process and we derive multi-type shape results related to its morphology.'
address: |
    Institut de Mathématiques\
    École Polytechinique Fédérale de Lausanne\
    CH-1015 Lausanne\
    Switzerland\
author:
- 'Leandro P. R. Pimentel'
title: 'Multi-type shape theorems for FPP models'
---

Introduction {#int}
============

In standard planar first-passage percolation [@HW65] each pair $\x$ and $\y$ of nearest-neighbor of $\bZ^2$ has an edge connecting them and each edge is equipped with a non-negative random variable (passage time) which may be interpreted as the time it takes for an infection to be transmitted from $\x$ to $\y$. We assume these random variables are i.i.d. with a continuous distribution $\bF$. The passage time $t(\gamma)$ for a nearest-neighbor path $\gamma$ is simply the sum of the passage times along the path. For $\x,\y\in\bZ^2$, the first-passage time from $\x$ to $\y$, which we denote $T(\x,\y)$, is the infimum of $t(\gamma)$ over all paths $\gamma$ from $\x$ to $\y$. For $t\geq 0$, let $\B(t)$ be the set of sites $\x$ reached from the origin $\0$ by time $t$, 