Universality for first passage percolation on sparse random graphs

• Shankar Bhamidi ^[3] ; Remco van der Hofstad ^[1] ; Gerard Hooghiemstra ^[2]
  1. [1] Eindhoven University of Technology

     Eindhoven University of Technology

     Países Bajos

     
  2. [2] Delft University of Technology

     Delft University of Technology

     Países Bajos

     
  3. [3] University of North Carolina (USA)

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 4, 2017, págs. 2568-2630
• Idioma: inglés
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• Resumen

  • We consider first passage percolation on the configuration model with nn vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X2logXX2log⁡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 LnLn for the weight of the optimal path, we show that Ln−(logn)/αnLn−(log⁡n)/αn converges to a limiting random variable, for some sequence αnαn. Furthermore, the hopcount satisfies a central limit theorem (CLT) with asymptotic mean and variance of order lognlog⁡n. The sequence αnαn 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 Ln−(logn)/αnLn−(log⁡n)/αn 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.