November 2023 Multisource invasion percolation on the complete graph
Louigi Addario-Berry, Jordan Barrett
Author Affiliations +
Ann. Probab. 51(6): 2131-2157 (November 2023). DOI: 10.1214/23-AOP1641


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)/n1/30 then Mn/n1 in probability. In this paper, we prove a complementary result: if k(n)/n1/3, then Mn/n0 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.

Funding Statement

During the preparation of this research, LAB was supported by an NSERC Discovery Grant and a Simons Fellowship in Mathematics.


The authors thank Ross Kang for pointing out the paper [22], and additionally thank the anonymous referees for comments, which improved the presentation of the paper.


Download Citation

Louigi Addario-Berry. Jordan Barrett. "Multisource invasion percolation on the complete graph." Ann. Probab. 51 (6) 2131 - 2157, November 2023.


Received: 1 August 2022; Revised: 1 May 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1641

Primary: 60K35
Secondary: 05C80 , 60C05 , 82B43 , 82C43

Keywords: Critical percolation , Erdős–Rényi random graph , Invasion percolation , minimum spanning trees

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • November 2023
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