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November 2013 Scaling window for mean-field percolation of averages
Jian Ding
Ann. Probab. 41(6): 4407-4427 (November 2013). DOI: 10.1214/12-AOP765

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.

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Jian Ding. "Scaling window for mean-field percolation of averages." Ann. Probab. 41 (6) 4407 - 4427, November 2013. https://doi.org/10.1214/12-AOP765

Information

Published: November 2013
First available in Project Euclid: 20 November 2013

zbMATH: 1291.60016
MathSciNet: MR3161479
Digital Object Identifier: 10.1214/12-AOP765

Subjects:
Primary: 60C05 , 60G70

Keywords: percolation , scaling window , stochastic distance model

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 6 • November 2013
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