Geodesic
The lower tail of the random minimum spanning tree
The electronic journal of combinatorics, Tome 14 (2007)
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Consider a complete graph $K_n$ where the edges have costs given by independent random variables, each distributed uniformly between 0 and 1. The cost of the minimum spanning tree in this graph is a random variable which has been the subject of much study. This note considers the large deviation probability of this random variable. Previous work has shown that the log-probability of deviation by $\varepsilon$ is $-\Omega(n)$, and that for the log-probability of $Z$ exceeding $\zeta(3)$ this bound is correct; $\log {\rm Pr}[Z \geq \zeta(3) + \varepsilon] = -\Theta(n)$. The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear, $\log {\rm Pr}[Z \leq \zeta(3) - \varepsilon] = -\Theta(n)$.
DOI : 10.37236/1004
Classification : 05C80, 60C05
Mots-clés : random variable, deviation probability, log-probability 
@article{10_37236_1004,
     author = {Abraham D. Flaxman},
     title = {The lower tail of the random minimum spanning tree},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/1004},
     zbl = {1113.05090},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1004/}
}
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Abraham D. Flaxman. The lower tail of the random minimum spanning tree. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/1004

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