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End date: 2019 × Type: Text × Subject: gelation × Subject: phase transition × WGL Institute: WIAS ×

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    A large-deviations approach to gelation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Andreis, Luisa; König, Wolfgang; Patterson, Robert
    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 Erdos-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 Erdos-Rényi graphs are connected.
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    Bilinear coagulation equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heydecker, Daniel; Patterson, Robert I.A.
    We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time tg ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.