tailieunhanh - Báo cáo toán học: "The lower tail of the random minimum spanning tree"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The lower tail of the random minimum spanning tree. | The lower tail of the random minimum spanning tree Abraham D. Flaxman Microsoft Research Redmond WA USA abie@ Submitted Sep 11 2006 Accepted Dec 15 2006 Published Jan 17 2007 Mathematics Subject Classifications 05C80 60C05 Abstract Consider a complete graph Kn 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 is Q n and that for the log-probability of Z exceeding 3 this bound is correct logPr Z 3 0 n . The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear logPr Z 3 0 n . 1 Introduction If the edge costs of the complete graph Kn are independent random variables each uniformly distributed between 0 and 1 then the cost of a minimum spanning tree is a random variable which has expectation asymptotically equal to 3 V1 i-3 6 . Furthermore after an appropriate rescaling this random variable converges in distribution to a Gaussian distribution with an explicitly known variance of about 8 . This note considers the large deviation probability of this random variable denoted Zn. In 9 as an example application of Talagrand s Inequality McDiarmid shows that Zn satisfies an exponential tail inequality of the form Pr Zn - 3 1 e-C-n. See also 4 for an alternative approach with additional details . Simple considerations show that for the log-probability of Zn exceeding 3 this bound is correct which is to say that logPr Zn 3 0 n . For example the probability that every edge incident to vertex 1 has cost at least 1 2 is 1 2 n-1 and conditioned on this event whp Zn 1 o 1 3 1 2 . THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N3 1 The behavior of the lower tail is not as simple to identify. A casual .

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