Showing papers by "Mathew D. Penrose published in 1997"
TL;DR: For n points placed uniformly at random on the unit square, it is known that the distribution of the minimal spanning tree on these points converges weakly to the double exponential as mentioned in this paper.
Abstract: For n points placed uniformly at random on the unit square, suppose $M_n$ (respectively, $M'_n$) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of $n \pi M_n^2 - \log n$ converges weakly to the double exponential; we give a new proof of this. We show that $P[M'_n = M_n] \to 1$, so that the same weak convergence holds for $M'_n$ .
528 citations
TL;DR: In this article, the critical value of λ, above which an infinite cluster exists a.s., is asymptotic to (∫ R d g (| x |) dx ) −1 as d → ∞.
33 citations