WIAS Preprint No. 2568, (2019)

A large-deviations approach to gelation


  • Andreis, Luisa
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Patterson, Robert I. A.
    ORCID: 0000-0002-3583-2857

2010 Mathematics Subject Classification

  • 05C80 60F10 60K35 82B26


  • Coagulation process, multiplicative coalescent, gelation, phase transition, large deviations, Erdős-Rényi random graph




A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected.

Appeared in

  • Random Structures and Algorithms (online first on 05.04.2021), DOI 10.1002/rsa.21007 under the new title ''A large-deviations principle for all the cluster sizes of a sparse Erdős-Rényi random graph ".

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