Percolation of averages in the stochastic mean field model: the near-supercritical regime

TL;DR

This paper investigates the length of the longest path with average weight constraints in a complete graph, focusing on the near-supercritical regime where the average weight threshold is just above a critical value, revealing exponential scaling behavior.

Contribution

It provides rigorous bounds on the path length in the near-supercritical regime, correcting previous non-rigorous predictions by Aldous.

Findings

Path length scales as $n e^{-C^*/ oot{ ext{eta}}}$ in the near-supercritical regime.

Establishes bounds with high probability for the path length.

Refines understanding of percolation thresholds in stochastic mean field models.

Abstract

For a complete graph of size $n$, assign each edge an i.i.d.\ exponential variable with mean $n$. For $\lambda>0$, consider the length of the longest path whose average weight is at most $\lambda$. It was shown by Aldous (1998) that the length is of order $\log n$ for $\lambda < 1/\mathrm{e}$ and of order $n$ for $\lambda > 1/\mathrm{e}$. In this paper, we study the near-supercritical regime where $\lambda = \mathrm{e}^{-1} +\eta$ with $\eta>0$ a small fixed number. We show that there exist two absolute constants $c^*, C^*>0$ such that with high probability the length is in between $n \mathrm{e}^{-C^*/\sqrt{\eta}}$ and $n \mathrm{e}^{-c^*/\sqrt{\eta}}$. Our result corrects a non-rigorous prediction of Aldous (2005).