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Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence

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In Bernardin et al. (Arch Ration Mech Anal 220(2):505–542, 2016) it has been proved that a linear Hamiltonian lattice field with two conservation laws, perturbed by a conservative stochastic noise, belongs to the \({\frac32}\)-Lévy/Diffusive universality class in the nonlinear fluctuating theory terminology (Spohn in J Stat Phys 154(5):1191–1227, 2014), i.e. energy superdiffuses like an asymmetric stable \({\frac32}\)-Lévy process and volume diffuses like a Brownian motion. According to this theory this should remain valid at zero tension if the harmonic potential is replaced by an even potential. In this work we consider a quartic anharmonicity and show that the result obtained in the harmonic case persists up to some small critical value of the anharmonicity.

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Acknowledgements

This work benefited from the support of the project EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) and of the PHC Pessoa Project 37854WM, and also in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program Non-equilibrium statistical physics (Code: ICTS/Prog-NESP/2015/10). C.B. thanks the French National Research Agency (ANR) for its support through the Grant ANR-15-CE40-0020-01 (LSD). P.G. thanks FCT/Portugal for support through the project UID/MAT/04459/2013. M.J. thanks CNPq for its support through the Grant 401628/2012-4 and FAPERJ for its support through the Grant JCNE E17/2012. M.J. was partially supported by NWO Gravitation Grant 024.002.003-NETWORKS. M.S. thanks the Labex CEMPI (ANR-11-LABX-0007-01) for its support. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (Grant Agreement No. 715734).

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Authors and Affiliations

  1. Université Côte d’Azur, CNRS, LJAD, Parc Valrose, 06108, NICE Cedex 02, France

    Cédric Bernardin

  2. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, no. 1, 1049-001, Lisboa, Portugal

    Patrícia Gonçalves

  3. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio De Janeiro, 22460-320, Brazil

    Milton Jara

  4. Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000, Lille, France

    Marielle Simon

Authors
  1. Cédric Bernardin
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  2. Patrícia Gonçalves
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  3. Milton Jara
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  4. Marielle Simon
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Correspondence to Cédric Bernardin.

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Communicated by H. Spohn

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Bernardin, C., Gonçalves, P., Jara, M. et al. Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence. Commun. Math. Phys. 361, 605–659 (2018). https://doi.org/10.1007/s00220-018-3191-z

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