Local limits of random spanning trees in random environment
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911691148623872 |
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| author | Makowiec, Luca |
| author_facet | Makowiec, Luca |
| contents | We study the edge overlap and local limit of the random spanning tree in random environment (RSTRE) on the complete graph with $n$ vertices and weights given by $\exp(-βω_e)$ for $ω_e$ uniformly distributed on $[0,1]$. We show that for $β$ growing with $β= o(n/\log n)$, the edge overlap is $(1+o(1)) β$, while for $β$ much larger than $n \log^2 n$, the edge overlap is $(1-o(1))n$. Furthermore, there is a transition of the local limit around $β= n$. When $β= o(n/ \log n)$ the RSTRE locally converges to the same limit as the uniform spanning tree, whereas for $β$ larger than $n \log^λn$, where $λ= λ(n) \rightarrow \infty$ arbitrarily slowly, the local limit of the RSTRE is the same as that of the minimum spanning tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local limits of random spanning trees in random environment Makowiec, Luca Probability Combinatorics 60K35 (Primary) 82B41, 82B44, 05C05 (Secondary) We study the edge overlap and local limit of the random spanning tree in random environment (RSTRE) on the complete graph with $n$ vertices and weights given by $\exp(-βω_e)$ for $ω_e$ uniformly distributed on $[0,1]$. We show that for $β$ growing with $β= o(n/\log n)$, the edge overlap is $(1+o(1)) β$, while for $β$ much larger than $n \log^2 n$, the edge overlap is $(1-o(1))n$. Furthermore, there is a transition of the local limit around $β= n$. When $β= o(n/ \log n)$ the RSTRE locally converges to the same limit as the uniform spanning tree, whereas for $β$ larger than $n \log^λn$, where $λ= λ(n) \rightarrow \infty$ arbitrarily slowly, the local limit of the RSTRE is the same as that of the minimum spanning tree. |
| title | Local limits of random spanning trees in random environment |
| topic | Probability Combinatorics 60K35 (Primary) 82B41, 82B44, 05C05 (Secondary) |
| url | https://arxiv.org/abs/2410.16836 |