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Noise sensitivity of the minimum spanning tree of the complete graph
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 6 / November 2024
- Published online by Cambridge University Press:
- 23 May 2024, pp. 708-723
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We study the noise sensitivity of the minimum spanning tree (MST) of the
$n$-vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by 
$n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov–Hausdorff–Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability 
$\varepsilon \gg n^{-1/3}$, then the pair of rescaled minimum spanning trees – before and after the noise – converges in distribution to independent random spaces. Conversely, if 
$\varepsilon \ll n^{-1/3}$, the GHP distance between the rescaled trees goes to 
$0$ in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of 
$n^{-1/3}$ coincides with the critical window of the Erdős-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.
