Supercritical minimum mean-weight cycles

TL;DR

This paper analyzes the supercritical regime of the minimum mean-weight cycle in a complete graph with exponential edge weights, revealing its asymptotic weight and cycle length as the number of vertices grows large.

Contribution

It provides the first detailed asymptotic characterization of the minimum mean-weight cycle in the supercritical regime, extending previous subcritical analysis.

Findings

Minimum mean weight asymptotically $(n e)^{-1}[1 + rac{ ext{pi}^2}{2 ext{log}^2 n}]$

Cycle length on the order of $( ext{log} n)^3$ in the supercritical regime

Completes the understanding of the phase transition in the model

Abstract

We study the weight and length of the minimum mean-weight cycle in the stochastic mean-field distance model, i.e., in the complete graph on $n$ vertices with edges weighted by independent exponential random variables. Mathieu and Wilson showed that the minimum mean-weight cycle exhibits one of two distinct behaviors, according to whether its mean weight is smaller or larger than $1/(ne)$; and that both scenarios occur with positive probability in the limit $n\to\infty$. If the mean weight is $< 1/(ne)$, the length is of constant order. If the mean weight is $> 1/(ne)$, it is concentrated just above $1/(n e)$, and the length diverges with $n$. The analysis of Mathieu--Wilson gives a detailed characterization of the subcritical regime, including the (non-degenerate) limiting distributions of the weight and length, but leaves open the supercritical behavior. We determine the asymptotics for the supercritical regime, showing that with high probability, the minimum mean weight is $(n e)^{-1}[1 + \pi^2/(2 \log^2 n) + O((\log n)^{-3})]$, and the cycle achieving this minimum has length on the order of $(\log n)^3$.