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Theory of Stochastic Laplacian Growth

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Abstract

We generalize the diffusion-limited aggregation by issuing many randomly-walking particles, which stick to a cluster at the discrete time unit providing its growth. Using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic Laplacian growth equation. A potential-theoretical analysis of the growth probabilities reveals connections with the tau-function of the integrable dispersionless limit of the two-dimensional Toda hierarchy, normal matrix ensembles, and the two-dimensional Dyson gas confined in a non-uniform magnetic field. We introduce the time-dependent Hamiltonian, which generates transitions between different classes of equivalence of closed curves, and prove the Hamiltonian structure of the interface dynamics. Finally, we propose a relation between probabilities of growth scenarios and the semi-classical limit of certain correlation functions of “light” exponential operators in the Liouville conformal field theory on a pseudosphere.

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Notes

  1. The potential equals zero on it

  2. It is supposed that the distant source is located at infinity.

  3. In fact, it becomes a cut-off dependent constant.

  4. In what follows we set \(M=1\) for notation simplicity. However, the construction of the Hamiltonian system can be performed for general M as well

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Authors and Affiliations

  1. International Institute of Physics, Federal University of Rio Grande do Norte, Natal, 59078-970, Brazil

    Oleg Alekseev & Mark Mineev-Weinstein

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  1. Oleg Alekseev
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  2. Mark Mineev-Weinstein
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Correspondence to Oleg Alekseev.

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Alekseev, O., Mineev-Weinstein, M. Theory of Stochastic Laplacian Growth. J Stat Phys 168, 68–91 (2017). https://doi.org/10.1007/s10955-017-1796-9

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  • DOI: https://doi.org/10.1007/s10955-017-1796-9

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