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Polymers and random graphs

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Abstract

We establish a precise connection between gelation of polymers in Lushnikov's model and the emergence of the giant component in random graph theory. This is achieved by defining a modified version of the Erdös-Rényi process; when contracting to a polymer state space, this process becomes a discrete-time Markov chain embedded in Lushnikov's process. The asymptotic distribution of the number of transitions in Lushnikov's model is studied. A criterion for a general Markov chain to retain the Markov property under the grouping of states is derived. We obtain a noncombinatorial proof of a theorem of Erdös-Rényi type.

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Buffet, E., Pulé, J.V. Polymers and random graphs. J Stat Phys 64, 87–110 (1991). https://doi.org/10.1007/BF01057869

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Key words

  • Gelation of polymers
  • giant component of random graph
  • grouping of states in a Markov chain
  • Erdös-Rényi theorem