Competition between Growths Governed by Bernoulli Percolation

#### O. Garet, R. Marchand

2006, v.12, Issue 4, 695-734

ABSTRACT

We study a competition model on $Z^d$ where the two infections are driven by supercritical Bernoulli percolation processes with distinct parameters $p$ and $q$. We prove that, for any $q$, there exist at most countably many values of $p<\min{\{q,\overrightarrow{p_c}\}}$ such that coexistence can occur. As a key step, we show that the norm associated to the chemical distance in supercritical Bernoulli percolation is strictly decreasing on $(p_c, \overrightarrow{p_c})$.

Keywords: percolation,first-passage percolation,chemical distance,competition,random growth