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Formation of Facets for an Effective Model of Crystal Growth

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Sojourns in Probability Theory and Statistical Physics - I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 298))

Abstract

We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a \(2+1\) solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to “grow” through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.

D. Ioffe–The research was supported by Israeli Science Foundation grant 1723/14, by The Leverhulme Trust through International Network Grant Laplacians, Random Walks, Bose Gas, Quantum Spin Systems and by the Meitner Humboldt Award.

S. Shlosman–The research has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). Part of this work has been carried out at IITP RAS. The support of Russian Foundation for Sciences (project No. 14-50-00150) is gratefully acknowledged.

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Notes

  1. 1.

    Actually, main results hold even with faster decay than (3).

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Author information

Authors and Affiliations

  1. Faculty of IE&M, Technion, 32000, Haifa, Israel

    Dmitry Ioffe

  2. Skolkovo Institute of Science and Technology, Moscow, Russia

    Senya Shlosman

  3. Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332, 13288, Marseille, France

    Senya Shlosman

  4. Institute of the Information Transmission Problems, RAS, Moscow, Russia

    Senya Shlosman

Authors
  1. Dmitry Ioffe
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  2. Senya Shlosman
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Correspondence to Dmitry Ioffe .

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Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

We dedicate this paper to Chuck Newman on the occasion of his 70th birthday

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Ioffe, D., Shlosman, S. (2019). Formation of Facets for an Effective Model of Crystal Growth. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - I. Springer Proceedings in Mathematics & Statistics, vol 298. Springer, Singapore. https://doi.org/10.1007/978-981-15-0294-1_9

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