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

  • N. V. Antonov
  • P. I. KakinEmail author
  • N. M. Lebedev


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.


Renormalization group Critical scaling Self-organized criticality Stochastic growth Critical exponents 



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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of PhysicsSaint Petersburg State UniversitySaint PetersburgRussia
  2. 2.N.N. Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubnaRussia

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