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Static Approach to Renormalization Group Analysis of Stochastic Models with Spatially Quenched Noise

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Abstract

A new “static” renormalization group approach to stochastic models of fluctuating surfaces with spatially quenched noise is proposed in which only time-independent quantities are involved. As examples, quenched versions of the Kardar–Parisi–Zhang model and its Pavlik’s modification, the Hwa–Kardar model of self-organized criticality, and Pastor–Satorras–Rothman model of landscape erosion are studied. It is shown that the logarithmic dimension in the quenched models is shifted by two units upwards in comparison to their counterparts with white in-time noise. Possible scaling regimes associated with fixed points of the renormalization group equations are found and the critical exponents are derived to the leading order of the corresponding \(\varepsilon \) expansions. Some exact values and relations for these exponents are obtained.

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Notes

  1. In principle, the stochastic equation may have time-dependent solutions. In particular, if the function h(x) is chosen in the form \(h(x)=h_0 t + h_1(\mathbf{x})\) and the nonlinearity depends only on the gradients of the field, like for the KPZ model, the left hand side of Eq. (4) becomes independent on t and can be equated to the right hand side. However, here we are interested only in a stationary state of the system and work within diagrammatic perturbation theory; then the solution is unique and time-independent. We thank one of the referees for bringing our attention to this point.

  2. The necessary steps of renormalization group analysis are thoroughly discussed and justified in several works, see, e.g. [65] or [60], Ch. 5 and 6. For the sake of brevity, here we restrict ourselves to only the necessary information.

  3. Another possibility would be to assume that all the couplings \(g_n\) are of the same order. Then the expansion in the full set of \(g_n\)’s would correspond to the expansion in the interaction n(h) or, equivalently, in V(h). In a one-coupling model such two possibilities are equivalent; in the multi-coupling models the loop expansion is richer in the sense that it performs a partial (and in our case infinite) resummation of the plain expansion in the interaction.

    In this connection, the scalar electrodynamics should be mentioned: there, the charge e of the scalar particle and the constant g of the scalar quartic interaction are related as \(g \propto e^2\). In the \(\varepsilon = 4-d\) expansion they are of the same order: \(g \propto e^2 \simeq \varepsilon \); see, e.g., [111,112,113] and the references.

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Acknowledgements

We are thankful to C. Duclut for bringing the work [42] to our attention. We also thank L. Ts. Adzhemyan, N.M. Gulitskiy, M. Hnatich, M.V. Kompaniets, and M.Yu. Nalimov for fruitful discussions. We are also thankful to the referees for useful comments and suggestions. The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”, and by the RFBR according to the research project 18-32-00238 (all the results concerning the KPZ model in Sect. 3).

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  1. Department of Physics, Saint Petersburg State University, 7/9 Universitetskaya Naberezhnaya, Saint Petersburg, Russia, 199034

    N. V. Antonov & P. I. Kakin

  2. N.N. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia, 141980

    N. M. Lebedev

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  1. N. V. Antonov
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  2. P. I. Kakin
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  3. N. M. Lebedev
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Correspondence to P. I. Kakin.

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Antonov, N.V., Kakin, P.I. & Lebedev, N.M. Static Approach to Renormalization Group Analysis of Stochastic Models with Spatially Quenched Noise. J Stat Phys 178, 392–419 (2020). https://doi.org/10.1007/s10955-019-02436-8

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