Multisource invasion percolation on the complete graph
Abstract
We consider invasion percolation on the randomlyweighted 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 multisource invasion percolation with differentlysized 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.06509
 Bibcode:
 2022arXiv220806509A
 Keywords:

 Mathematics  Probability;
 Primary: 60K35;
 Secondary: 60C05;
 05C80;
 82B43;
 82C43
 EPrint:
 35 pages