TY - JOUR

T1 - Random walk on a random surface: implications of non-perturbative concepts and dynamical emergence of Galilean symmetry

AU - Антонов, Николай Викторович

AU - Гулицкий, Николай Михайлович

AU - Какинь, Полина Игоревна

AU - Романчук, Анастасия Сергеевна

PY - 2025/3/3

Y1 - 2025/3/3

N2 - We study a model of random walk on a fluctuating rough surface using the field-theoretic renormalization group (RG). The surface is modelled by the well-known Kardar-Parisi-Zhang (KPZ) stochastic equation while the random walk is described by the standard diffusion equation for a particle in a uniform gravitational field. In the RG approach, possible types of infrared (IR) asymptotic (long-time, large-distance) behaviour are determined by IR attractive fixed points. Within the one-loop RG calculation (the leading order in ϵ = 2 − d , d being the spatial dimension), we found six possible fixed points or curves of points. Two of them can be IR attractive: the Gaussian point (free theory) and the nontrivial point where the KPZ surface is rough but its interaction with the random walk is irrelevant. For those perturbative fixed points, the spreading law for a particles’ cloud coincides with that for ordinary random walk, R ( t ) ∼ t1/2 . We also explored consequences of the presumed existence in the KPZ model of a non-perturbative strong-coupling fixed point. We found that it gives rise to an IR attractive fixed point in the full-scale model with the nontrivial spreading law R ( t ) ∼ t 1/z , where the exponent z < 2 can be inferred from the non-perturbative analysis of the KPZ model. Thus, the spreading becomes faster on a rough fluctuating surface in comparison to a smooth one, the phenomenon known as superdiffusion. What is more, the Galilean-type symmetry inherent for the pure KPZ model extends dynamically to the IR asymptotic behaviour of the Green’s functions of the full model.

AB - We study a model of random walk on a fluctuating rough surface using the field-theoretic renormalization group (RG). The surface is modelled by the well-known Kardar-Parisi-Zhang (KPZ) stochastic equation while the random walk is described by the standard diffusion equation for a particle in a uniform gravitational field. In the RG approach, possible types of infrared (IR) asymptotic (long-time, large-distance) behaviour are determined by IR attractive fixed points. Within the one-loop RG calculation (the leading order in ϵ = 2 − d , d being the spatial dimension), we found six possible fixed points or curves of points. Two of them can be IR attractive: the Gaussian point (free theory) and the nontrivial point where the KPZ surface is rough but its interaction with the random walk is irrelevant. For those perturbative fixed points, the spreading law for a particles’ cloud coincides with that for ordinary random walk, R ( t ) ∼ t1/2 . We also explored consequences of the presumed existence in the KPZ model of a non-perturbative strong-coupling fixed point. We found that it gives rise to an IR attractive fixed point in the full-scale model with the nontrivial spreading law R ( t ) ∼ t 1/z , where the exponent z < 2 can be inferred from the non-perturbative analysis of the KPZ model. Thus, the spreading becomes faster on a rough fluctuating surface in comparison to a smooth one, the phenomenon known as superdiffusion. What is more, the Galilean-type symmetry inherent for the pure KPZ model extends dynamically to the IR asymptotic behaviour of the Green’s functions of the full model.

KW - disordered systems

KW - kinetic roughening

KW - non-equilibrium behavior

KW - random walks

KW - renormalization group

UR - https://www.mendeley.com/catalogue/c242a9a0-7c8e-30e3-aaa0-d68784fc7238/

U2 - 10.1088/1751-8121/adbc51

DO - 10.1088/1751-8121/adbc51

M3 - Article

VL - 58

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 11

M1 - 115001

ER -