Summary: We consider invasion percolation on the complete graph \(K_n\), 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 \(M_n\) be the size of the largest tree in this forest. Logan, Molloy and Pralat (2018) proved that if \(k(n) / n^{1/3} \to 0\) then \(M_n / n \to 1\) in probability. In this paper, we prove a complementary result: if \(k(n) / n^{1/3} \to \infty\), then \(M_n / n \to 0\) in probability. This establishes the existence of a phase transition in the structure of the invasion percolation forest around \(k(n) \asymp n^{1/3}\).
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.