Union of Random Trees and Applications
Abstract
In 1986, Janson showed that the number of edges in the union of $k$ random spanning trees in the complete graph $K_n$ is a shifted Poisson distribution. Using results from the theory of electrical networks, we provide a new proof of this result, and we obtain an explicit rate of convergence. This rate of convergence allows us to show a new upper tail bound on the number of trees in $G(n,p)$, for $p$ a constant not depending on $n$. The number of edges in the union of $k$ random trees is related to moments of the number of spanning trees in $G(n, p)$. As an application, we prove the law of the iterated logarithm for the number of spanning trees in $G(n,p)$. More precisely, consider the infinite random graph $G(\mathbb{N}, p)$, with vertex set $\mathbb{N}$ and where each edge appears independently with constant probability $p$. By restricting to $\{1, 2, \dotsc, n\}$, we obtain a series of nested Erdös-Réyni random graphs $G(n,p)$. We show that a scaled version of the number of spanning trees satisfies the law of the iterated logarithm.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.06208
- arXiv:
- arXiv:1701.06208
- Bibcode:
- 2017arXiv170106208J
- Keywords:
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- Mathematics - Combinatorics