Repeat times and a two-weight UST model
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908732852535296 |
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| author | De Ambroggio, Umberto Makowiec, Luca |
| author_facet | De Ambroggio, Umberto Makowiec, Luca |
| contents | We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+γ}$ with probability $1/n$ and $1$ otherwise. Whenever $γ$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1/3} \log n$, up to a $\log \log n$ correction. Our approach uses estimates on repeat times for selecting components in a critical Erdős-Rényi graph, as well as concentration bounds on the sums of diameters of these components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21977 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Repeat times and a two-weight UST model De Ambroggio, Umberto Makowiec, Luca Probability Combinatorics 60K35 (Primary) 82B41, 82B44, 05C05 (Secondary) We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+γ}$ with probability $1/n$ and $1$ otherwise. Whenever $γ$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1/3} \log n$, up to a $\log \log n$ correction. Our approach uses estimates on repeat times for selecting components in a critical Erdős-Rényi graph, as well as concentration bounds on the sums of diameters of these components. |
| title | Repeat times and a two-weight UST model |
| topic | Probability Combinatorics 60K35 (Primary) 82B41, 82B44, 05C05 (Secondary) |
| url | https://arxiv.org/abs/2512.21977 |