# Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model

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## Abstract

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## 1. Introduction

## 2. Description of the Model

## 3. Field-Theoretic Formulation and Renormalization of the Model

## 4. RG Equations, RG Functions, and Fixed Points

## 5. Critical Dimensions and Scaling Behavior

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Here and below, the subscript 0 refers to bare parameters which will be renormalized in the following. |

2 | Detailed discussion of fractional derivatives can be found in [21]. |

3 | Although ${u}_{0}$ is not an expansion parameter in perturbation theory, its renormalized counterpart is dimensionless, enters into renormalization constants and RG functions, and should be treated on equal footing with ${g}_{0}$. We also recall that ${\lambda}_{0}=1$. |

## References

- Edwards, S.F.; Wilkinson, D.R. The Surface Statistics of a Granular Aggregate. Proc. R. Soc.
**1982**, 381, 17–31. [Google Scholar] - Kardar, M.; Parisi, G.; Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett.
**1986**, 56, 889–892. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yan, H.; Kessler, D.A.; Sander, L.M. Roughening phase transition in surface growth. Phys. Rev. Lett.
**1990**, 64, 926–929. [Google Scholar] [CrossRef] [PubMed] - Yan, H.; Kessler, D.A.; Sander, L.M. Kinetic Roughening in Surface Growth. MRS Online Proc. Libr.
**1992**, 278, 237–247. [Google Scholar] [CrossRef] - Pavlik, S.I. Scaling for a growing phase boundary with nonlinear diffusion. JETP
**1994**, 79, 303, [Translated from the Russian: ZhETF**1994**, 106, 553.]. [Google Scholar] - Halpin-Healy, T.; Zhang, Y.-C. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Phys. Rep.
**1995**, 254, 215–414. [Google Scholar] [CrossRef] - Barabási, A.-L.; Stanley, H.E. Fractal Concepts in Surface Growth; Cambridge University Press: Cambridge, MA, USA, 1995. [Google Scholar]
- Antonov, N.V.; Vasil’ev, A.N. The quantum-field renormalization group in the problem of a growing phase boundary. JETP
**1995**, 81, 485, [Translated from the Russian: ZhETF**1995**, 108, 885.]. [Google Scholar] - Pastor-Satorras, R.; Rothman, D.H. Stochastic equation for the erosion of inclined topography. Phys. Rev. Lett.
**1998**, 80, 4349–4352. [Google Scholar] [CrossRef] [Green Version] - Pastor-Satorras, R.; Rothman, D.H. Scaling of a slope: The erosion of tilted landscapes. J. Stat. Phys.
**1998**, 93, 477–500. [Google Scholar] [CrossRef] - Antonov, N.V.; Kakin, P.I. Scaling in erosion of landscapes: Renormalization group analysis of a model with infinitely many couplings. Theor. Math. Phys.
**2017**, 190, 193–203. [Google Scholar] [CrossRef] [Green Version] - Duclut, C.; Delamotte, B. Nonuniversality in the erosion of tilted landscapes. Phys. Rev. E
**2017**, 96, 012149. [Google Scholar] [CrossRef] [Green Version] - Song, T.; Xia, H. Kinetic roughening and nontrivial scaling in the Kardar–Parisi–Zhang growth with long-range temporal correlations. J. Stat. Mech.
**2021**, 2021, 073203. [Google Scholar] [CrossRef] - Marinari, E.; Parisi, G.; Ruelle, D.; Windey, P. Random Walk in a Random Environment and 1/f Noise. Phys. Rev. Lett.
**1983**, 50, 1223. [Google Scholar] [CrossRef] - Marinari, E.; Parisi, G.; Ruelle, D.; Windey, P. On the interpretation of 1/f noise. Commun. Math. Phys.
**1983**, 89, 1–12. [Google Scholar] [CrossRef] - Fisher, D.S. Random walks in random environments. Phys. Rev. A
**1984**, 30, 960. [Google Scholar] [CrossRef] - Fisher, D.S.; Friedan, D.; Qiu, Z.; Shenker, S.J.; Shenker, S.H. Random walks in two-dimensional random environments with constrained drift forces. Phys. Rev. A
**1985**, 31, 3841–3845. [Google Scholar] [CrossRef] - Kravtsov, V.E.; Lerner, I.V.; Yudson, V.I. The Einstein relation and exact Gell-Mann-Low function for random walks in media with random drifts. Phys. Lett. A
**1986**, 119, 203–206. [Google Scholar] [CrossRef] - Honkonen, J.; Pis’mak, Y.M.; Vasil’ev, A.N. Zero beta function for a model of diffusion in potential random field. J. Phys. A Math. Gen.
**1988**, 21, L835–L841. [Google Scholar] [CrossRef] - Bouchaud, J.-P.; Georges, A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep.
**1990**, 195, 127–293. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef] - Zeitouni, O. Random walks in random environment. In Computational Complexity; Meyers, R., Ed.; Springer: New York, NY, USA, 2012. [Google Scholar]
- Révész, P. Random Walk in Random and Non-Random Environments, 3rd ed.; World Scientific Book: Singapore, 2013. [Google Scholar]
- Haldar, A.; Basu, A. Marching on a rugged landscape: Universality in disordered asymmetric exclusion processes. Phys. Rev. Res.
**2020**, 2, 043073. [Google Scholar] [CrossRef] - Hairer, M. Solving the KPZ equation. Ann. Math.
**2013**, 178, 559–664. [Google Scholar] [CrossRef] [Green Version] - Hairer, M.; Shen, H. A central limit theorem for the KPZ equation. arXiv
**2016**, arXiv:1507.01237. [Google Scholar] [CrossRef] [Green Version] - Hairer, M. Exactly solving the KPZ equation. In Random Growth Models. Proceedings of Symposia in Applied Mathematics; Damron, M., Rassoul-Agha, F., Seppäläinen, T., Damron, M., Rassoul-Agha, F., Seppäläinen, T., Eds.; American Mathematical Society: Providence, RI, USA, 2018; Volume 75, p. 75. [Google Scholar]
- Corwin, I.; Shen, H. Some recent progress in singular stochastic partial differential equations. Bull. Am. Math. Soc.
**2020**, 57, 409–454. [Google Scholar] [CrossRef] [Green Version] - Barraquand, G.; Corwin, I. Stationary measures for the log-gamma polymer and KPZ equation in half-space. arXiv
**2022**, arXiv:2203.11037. [Google Scholar] - Falkovich, G.; Gawȩdzki, K.; Vergassola, M. Particles and fields in fluid turbulence. Rev. Mod. Phys.
**2001**, 73, 913–975. [Google Scholar] [CrossRef] - Pruessner, G. Self-Organized Criticality: Theory, Models and Characterisation; Cambridge University Press: Cambridge, MA, USA, 2012. [Google Scholar]
- Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett.
**1987**, 59, 381–384. [Google Scholar] [CrossRef] - Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality. Phys. Rev. A
**1988**, 38, 364–374. [Google Scholar] [CrossRef] - Maslov, S.; Tang, C.; Zhang, Y.-C. 1/f Noise in Bak-Tang-Wiesenfeld Models on Narrow Stripes. Phys. Rev. Lett.
**1999**, 83, 2449–2452. [Google Scholar] [CrossRef] [Green Version] - Vasiliev, A.N. The Field Theoretic Renormalization Group in Critical Behaviour Theory and Stochastic Dynamics; Chapman & Hall/CRC: Boca Raton, FL, USA, 2004; [Translated from the Russian: St Petersburg Institute of Nuclear Physics: Gatchina, Russia, 1999; ISBN 5-86763-122-2. [Google Scholar]
- Adzhemyan, L.T.; Antonov, N.V.; Vasil’ev, A.N. The Field Theoretic Renormalization Group in Fully Developed Turbulence; Gordon & Breach: London, UK, 1999. [Google Scholar]
- Antonov, N.V. Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field. Phys. Rev. E
**1999**, 60, 6691–6707. [Google Scholar] [CrossRef] [Green Version] - Adzhemyan, L.T.; Antonov, N.V.; Vasil’ev, A.N. Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar. Phys. Rev. E
**1998**, 58, 1823–1835. [Google Scholar] [CrossRef] [Green Version] - Avellaneda, M.; Majda, A. Mathematical models with exact renormalization for turbulent transport II: Non-Gaussian statistics, fractal interfaces, and the sweeping effect. Commun. Math. Phys.
**1992**, 146, 139–204. [Google Scholar] [CrossRef] - Hwa, T.; Kardar, M. Dissipative transport in open systems: An investigation of self-organized criticality. Phys. Rev. Lett.
**1989**, 62, 1813–1816. [Google Scholar] [CrossRef] - Hwa, T.; Kardar, M. Avalanches, hydrodynamics and great events in models of sandpiles. Phys. Rev. A
**1992**, 45, 7002–7023. [Google Scholar] [CrossRef]

F | ${\mathit{\theta}}^{\prime}\mathit{\theta}$ | ${\mathit{h}}^{\prime}$ | h | ${\mathit{\nu}}_{0}$, $\mathit{\nu}$ | ${\mathit{g}}_{0}$ | ${\mathit{u}}_{0}$ | g, u | $\mathit{\mu}$, m |
---|---|---|---|---|---|---|---|---|

${d}_{F}^{k}$ | d | $d+2$ | $-2$ | $-2$ | $\epsilon $ | $\eta $ | 0 | 1 |

${d}_{F}^{\omega}$ | 0 | $-1$ | 1 | 1 | 0 | 0 | 0 | 0 |

${d}_{F}$ | d | d | 0 | 0 | $\epsilon $ | $\eta $ | 0 | 1 |

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**MDPI and ACS Style**

Antonov, N.V.; Gulitskiy, N.M.; Kakin, P.I.; Kerbitskiy, D.A.
Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model. *Universe* **2023**, *9*, 139.
https://doi.org/10.3390/universe9030139

**AMA Style**

Antonov NV, Gulitskiy NM, Kakin PI, Kerbitskiy DA.
Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model. *Universe*. 2023; 9(3):139.
https://doi.org/10.3390/universe9030139

**Chicago/Turabian Style**

Antonov, Nikolay V., Nikolay M. Gulitskiy, Polina I. Kakin, and Dmitriy A. Kerbitskiy.
2023. "Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model" *Universe* 9, no. 3: 139.
https://doi.org/10.3390/universe9030139