In this talk we will consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy roughly N^alpha vertices, for some 0 < alpha < 1. The value of alpha is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. These results are in sharp contrast with the picture in the lattice case.
This is a joint work with Yael Dekel, Elchanan Mossel and Yuval Peres.