Multisource invasion percolation on the complete graph

Abstract

We consider invasion percolation on the complete graph Kn, started from some number k(n) of distinct source vertices. The outcome of the process is a forest consisting of k(n) trees, each containing exactly one source. Let Mn be the size of the largest tree in this forest. Logan, Molloy and Pralat (2018) proved that if k(n)/n1/3→0 then Mn/n→1 in probability. In this paper, we prove a complementary result: if k(n)/n1/3→∞, then Mn/n→0 in probability. This establishes the existence of a phase transition in the structure of the invasion percolation forest around k(n)≍n1/3. Our arguments rely on the connection between invasion percolation and critical percolation, and on a coupling between multisource invasion percolation with differently-sized source sets. A substantial part of the proof is devoted to showing that, with high probability, a certain fragmentation process on large random binary trees leaves no components of macroscopic size.