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

Abstract

For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $\lambda \gt 0$, 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 $\lambda \lt 1/\mathrm{e}$ and of order $n$ for $\lambda \gt 1/\mathrm{e}$. In this paper, we study the near-supercritical regime where $\lambda = \mathrm{e}^{-1} +\eta$ with $\eta \gt 0$ a small fixed number. We show that there exist two absolute constants $C_1, C_2 \gt 0$ such that with high probability the length is in between $n \mathrm{e}^{-C_1/\sqrt{\eta}}$ and $n \mathrm{e}^{-C_2/\sqrt{\eta}}$. Our result corrects a non-rigorous prediction of Aldous (2005).