Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.07357 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915025442045952 |
|---|---|
| author | Israeli, Omer Peled, Yuval |
| author_facet | Israeli, Omer Peled, Yuval |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07357 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Noise Sensitivity of the Minimum Spanning Tree of the Complete Graph Israeli, Omer Peled, Yuval Probability Combinatorics 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. |
| title | Noise Sensitivity of the Minimum Spanning Tree of the Complete Graph |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2306.07357 |