Multi-source invasion percolation on the complete graph

TL;DR

This paper studies invasion percolation on complete graphs with multiple sources, revealing a phase transition in the size of the largest tree depending on the number of sources relative to the graph size.

Contribution

It establishes the critical threshold for the size of the largest tree in multi-source invasion percolation on complete graphs, complementing previous results and identifying a phase transition.

Findings

If $k(n)/n^{1/3} o 0$, the largest tree occupies almost the entire graph.

If $k(n)/n^{1/3} o \infty$, the largest tree becomes negligible.

A phase transition occurs around $k(n) hicksim n^{1/3}$.

Abstract

We consider invasion percolation on the randomly-weighted 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 (arXiv:1806.10975) 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 multi-source 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.