[Paper Review] Scaling window for mean-field percolation of averages

Jian Ding|arXiv (Cornell University)|Oct 14, 2011

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper resolves a long-standing question by Aldous on the critical behavior of mean-field percolation in a complete graph with i.i.d. exponential edge weights. It establishes a precise scaling window of order $(\log n)^{-2}$ around $\lambda = 1/e$, showing that the longest path with average weight at most $\lambda$ is of order $(\log n)^3$ within this window, and exhibits a second transition point beyond it, revealing richer critical dynamics than previously expected.

ABSTRACT

For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $λ>0$, consider the length of the longest path whose average weight is at most $λ$. It was shown by Aldous (1998) that the length is of order $\log n$ for $λ< 1/\mathrm{e}$ and of order $n$ for $λ> 1/\mathrm{e}$. Aldous (2003) posed the question on detailed behavior at and near criticality $1/\mathrm{e}$. In particular, Aldous asked whether there exist scaling exponents $μ, ν$ such that for $λ$ within $1/\mathrm{e}$ of order $n^{-μ}$, the length for the longest path of average weight at most $λ$ has order $n^ν$. We answer this question by showing that the critical behavior is far richer: For $λ$ around $1/\mathrm{e}$ within a window of $α(\log n)^{-2}$ with a small absolute constant $α>0$, the longest path is of order $(\log n)^3$. Furthermore, for $λ\geq 1/\mathrm{e} + β(\log n)^{-2}$ with $β$ 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 $1/\mathrm{e} + [α(\log n)^{-2}, β(\log n)^{-2}]$. In addition, we demonstrate a smooth transition from subcritical to critical regime. Our results were not known before even in a heuristic sense.

Motivation & Objective

• To resolve Aldous' open question on the detailed behavior of the longest path with average weight at most $\lambda$ near the critical threshold $\lambda = 1/e$ in a mean-field random graph.
• To determine the correct scaling window for the phase transition in the mean-field percolation model with exponential edge weights.
• To establish the existence of a second transition point in the interval $[1/e + \alpha(\log n)^{-2}, 1/e + \beta(\log n)^{-2}]$ beyond the critical window.
• To demonstrate a smooth transition from the subcritical to critical regime, quantifying the behavior of the path length as $\lambda$ approaches $1/e$ from below.

Proposed method

• Uses a probabilistic analysis of paths in a complete graph with i.i.d. exponential edge weights of mean $n$, focusing on the longest path whose average weight is at most $\lambda$.
• Applies second moment methods and conditional probability estimates to prove the existence of long paths with high probability in the critical window.
• Employs a decomposition of paths into segments based on their length and weight, using conditional independence and moment bounds to control dependencies.
• Derives bounds on the expected number of long paths and their second moments to apply Chebyshev’s inequality and prove positive probability of existence.
• Introduces a refined analysis of path weights and maximum weights along paths, using exponential tail estimates and concentration inequalities.
• Demonstrates smooth interpolation from subcritical to critical behavior by analyzing the path length as $\lambda$ approaches $1/e$ from below.

Experimental results

Research questions

• RQ1 Is there a scaling window around $\lambda = 1/e$ such that the longest path with average weight at most $\lambda$ has length of order $(\log n)^3$?
• RQ2 Does the model exhibit a second phase transition in the interval $[1/e + \alpha(\log n)^{-2}, 1/e + \beta(\log n)^{-2}]$ for large absolute constants $\alpha, \beta$?
• RQ3 Can the transition from subcritical to critical behavior be shown to be smooth, with the path length scaling as $\Theta((1/e - \lambda)^{-1} \log n)$ for $\lambda < 1/e - (\log n)^{-2}$?
• RQ4 Is the critical window narrower than the $n^{-\mu}$ scaling proposed by Aldous, and does it scale as $(\log n)^{-2}$?
• RQ5 What is the exact order of magnitude of the longest path length at $\lambda = 1/e$, and does it satisfy $n^{o(1)}$ as conjectured by Aldous?

Key findings

• For $\lambda$ within $\alpha(\log n)^{-2}$ of $1/e$, the longest path with average weight at most $\lambda$ is of order $(\log n)^3$ with high probability.
• For $\lambda \geq 1/e + \beta(\log n)^{-2}$ with a large absolute constant $= \beta$, the longest path is at least of order $n^{1/4}$ with high probability.
• There exists a second phase transition in the interval $[1/e + \alpha(\log n)^{-2}, 1/e + \beta(\log n)^{-2}]$, indicating richer critical behavior than previously expected.
• For $\lambda < 1/e - (\log n)^{-2}$, the path length scales as $\Theta((1/e - \lambda)^{-1} \log n)$, demonstrating a smooth transition from subcritical to critical regime.
• The critical window is of order $(\log n)^{-2}$, confirming that the scaling is not of the form $n^{-\mu}$ as conjectured by Aldous.
• The result confirms Aldous' heuristic guess that $L(n, 1/e) = n^{o(1)}$, as $(\log n)^3$ is subpolynomial in $n$.