Average-weight percolation on the complete graph

TL;DR

This paper investigates the phase transition in the length of the longest average-weight path in a complete graph with exponential edge weights, pinpointing the critical point and providing precise asymptotics near it.

Contribution

It precisely locates the phase transition point and derives sharp asymptotics for the longest path length near criticality, using a novel exploration mechanism.

Findings

Phase transition occurs at p=1/e.

Longest path length is logarithmic below the transition.

Path length becomes polynomial above the transition.

Abstract

Attach to each edge of the complete graph on $n$ vertices, i.i.d. exponential random variables with mean $n$. Aldous [1] proved that the longest path with average weight below $p$ undergoes a phase transition at $p=\frac{1}{e}$: it is $o(n)$ when $p<\frac{1}{e}$ and of order $n$ if $p>\frac1e$. Later, Ding [4] revealed a finer phase transition around $\frac{1}{e}$: there exist $c'>c>0$ such that the length of the longest path is of order $\ln^3 n$ if $ p \le \frac{1}{e}+\frac{c}{\ln^2 n}$ and is polynomial if $p\ge \frac{1}{e}+\frac{c'}{\ln^2 n}$. We identify the location of this phase transition and obtain sharp asymptotics of the length near criticality. The proof uses an exploration mechanism mimicking a branching random walk with selection introduced by Brunet and Derrida [3].