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July 2017 Universality for first passage percolation on sparse random graphs
Shankar Bhamidi, Remco van der Hofstad, Gerard Hooghiemstra
Ann. Probab. 45(4): 2568-2630 (July 2017). DOI: 10.1214/16-AOP1120

Abstract

We consider first passage percolation on the configuration model with $n$ vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform $X^{2}\log{X}$-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path namely the hopcount.

Writing $L_{n}$ for the weight of the optimal path, we show that $L_{n}-(\log{n})/\alpha_{n}$ converges to a limiting random variable, for some sequence $\alpha_{n}$. Furthermore, the hopcount satisfies a central limit theorem (CLT) with asymptotic mean and variance of order $\log{n}$. The sequence $\alpha_{n}$ and the norming constants for the CLT are expressible in terms of the parameters of an associated continuous-time branching process that describes the growth of neighborhoods around a uniformly chosen vertex in the random graph. The limit of $L_{n}-(\log{n})/\alpha_{n}$ equals the sum of the logarithm of the product of two independent martingale limits, and a Gumbel random variable. So far, for sparse random graph models, such results have only been shown for the special case where the edge weights have an exponential distribution, wherein the Markov property of this distribution plays a crucial role in the technical analysis of the problem.

The proofs in the paper rely on a refined coupling between shortest path trees and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination.

Citation

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Shankar Bhamidi. Remco van der Hofstad. Gerard Hooghiemstra. "Universality for first passage percolation on sparse random graphs." Ann. Probab. 45 (4) 2568 - 2630, July 2017. https://doi.org/10.1214/16-AOP1120

Information

Received: 1 May 2014; Revised: 1 September 2015; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1376.60018
MathSciNet: MR3693970
Digital Object Identifier: 10.1214/16-AOP1120

Subjects:
Primary: 05C80, 60C05, 90B15

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 4 • July 2017
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