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January 2005 Stochastic processes in random graphs
Anatolii A. Puhalskii
Ann. Probab. 33(1): 337-412 (January 2005). DOI: 10.1214/009117904000000784

Abstract

We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint distribution of the number of connected components, of the sizes of the giant components and of the numbers of the excess edges of the giant components. For the supercritical case, we obtain the asymptotics of normal deviations and the logarithmic asymptotics of large and moderate deviations of the joint distribution of the number of components, of the size of the largest component and of the number of the excess edges of the largest component. For the critical case, we obtain the logarithmic asymptotics of moderate deviations of the joint distribution of the sizes of connected components and of the numbers of the excess edges. Some related asymptotics are also established. The proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory. As a byproduct of the results, we provide some additional insight into the nature of phase transitions in sparse random graphs.

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Anatolii A. Puhalskii. "Stochastic processes in random graphs." Ann. Probab. 33 (1) 337 - 412, January 2005. https://doi.org/10.1214/009117904000000784

Information

Published: January 2005
First available in Project Euclid: 11 February 2005

zbMATH: 1096.60008
MathSciNet: MR2118868
Digital Object Identifier: 10.1214/009117904000000784

Subjects:
Primary: 05C80
Secondary: 60C05 , 60F05 , 60F10 , 60F17

Keywords: connected components , idempotent probability , large deviation principle , large deviations , Moderate deviations , Phase transitions , Random graphs , Stochastic processes , weak convergence

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 1 • January 2005
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