Critical window for the configuration model: finite third moment degrees

Research output: Contribution to journalArticleAcademicpeer-review

11 Citations (Scopus)
121 Downloads (Pure)

Abstract

We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order n 2/3 and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.

Original languageEnglish
Article number16
Pages (from-to)1
Number of pages33
JournalElectronic Journal of Probability
Volume22
Issue number16
DOIs
Publication statusPublished - 2017

Keywords

  • Brownian excursions with parabolic drift
  • Critical configuration model
  • Finite third moment degree
  • Multiplicative coalescent
  • Scaling window
  • Universality

Fingerprint Dive into the research topics of 'Critical window for the configuration model: finite third moment degrees'. Together they form a unique fingerprint.

Cite this