Scaling window for mean-field percolation of averages

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 [Combin. Probab. Comput. 7 (1998) 1–10] that the length is of order $\log n$ for $\lambda<1/\mathrm{e}$ and of order $n$ for $\lambda>1/\mathrm{e}$. Aldous [Open problems (2003) Preprint] posed the question on detailed behavior at and near criticality $1/\mathrm{e}$. In particular, Aldous asked whether there exist scaling exponents $\mu$, $\nu$ such that for $\lambda$ within $1/\mathrm{e}$ of order $n^{-\mu}$, the length for the longest path of average weight at most $\lambda$ has order $n^{\nu}$. We answer this question by showing that the critical behavior is far richer: For $\lambda$ around $1/\mathrm{e}$ within a window of $\alpha(\log n)^{-2}$ with a small absolute constant $\alpha>0$, the longest path is of order $(\log n)^{3}$. Furthermore, for $\lambda\geq1/\mathrm{e}+\beta(\log n)^{-2}$ with $\beta$ a large absolute constant, the longest path is at least of length a polynomial in $n$. An interesting consequence of our result is the existence of a second transition point in $1/\mathrm{e}+[\alpha(\log n)^{-2},\beta(\log n)^{-2}]$. In addition, we demonstrate a smooth transition from subcritical to critical regime. Our results were not known before even in a heuristic sense.