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A Large Deviation Result on the Number of Small Subgraphs of a Random Graph

Published online by Cambridge University Press:  12 April 2001

VAN H. VU
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: vanhavu@@microsoft.com)

Abstract

Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.

[bull ] What is the upper tail probability Pr(YH [ges ] (1 + ε)[ ](YH))?

[bull ] For which λ does YH have sub-Gaussian behaviour, namely

(formula here)

where c is a positive constant?

[bull ] Fixing λ = ω(1) in advance, find a reasonably small tail T = T(λ) such that

(formula here)

We prove a general concentration result which contains a partial answer to each of these questions. The heart of the proof is a new martingale inequality, due to J. H. Kim and the present author [13].

Type
Research Article
Copyright
2001 Cambridge University Press

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