Description of KPZ interface growth by stochastic Loewner evolution

Abstract: In this study, we investigate the relationship between the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) equation and the stochastic Loewner equation (SLE), which is a one parameter family of the conformal mappings involving stochasticity. The author shows the correspondence between 1D KPZ equation with height function $h(x,t)=(3t<sup>2x+x<sup>3)/6t$ and Loewner equation driven by a nonlinear stochastic process, wherein the 1D dynamics of interface growth is characterized by Loewner entropy $S_{Loew}\simeq-\ln{t/κ}$. These results were numerically verified with discussions in relation to the universality in non-equilibrium statistical physics.

• The paper establishes a rigorous mapping between the 1D KPZ equation and a modified stochastic Loewner evolution framework.
• The paper validates KPZ universality scaling exponents and Loewner entropy behavior through both analytic derivations and numerical simulations.
• The paper demonstrates that Loewner entropy serves as a diagnostic classifier for universality classes in nonlinear stochastic growth.

SLE-Based Characterization of the 1D KPZ Equation

Introduction

This work rigorously interrogates the mathematical correspondence between one-dimensional Kardar-Parisi-Zhang (KPZ) interface growth and stochastic Loewner evolution (SLE). By formulating a mapping between the 1D KPZ equation and a modified Loewner equation driven by a nonlinear stochastic process, the study delivers a framework where the stochastic geometry of growing interfaces is captured via conformal mappings and Loewner entropy. The derivations are substantiated through analytic calculations and corroborated by numerical simulations that reproduce KPZ universality scaling exponents.

KPZ Equation and Universality

The KPZ equation

$$\frac{\partial h(x,t)}{\partial t} = \nu \frac{\partial^2 h(x,t)}{\partial x^2} + \frac{\lambda}{2} \left(\frac{\partial h(x,t)}{\partial x}\right)^2 + \sqrt{\kappa}\,\eta(t)$$

describes the evolution of a stochastic interface height $h(x,t)$ subject to nonlinear growth and noise. The familiar KPZ exponents $\alpha=1/2$, $\beta=1/3$, and $z=3/2$ define the roughness, growth, and dynamic scaling relations, respectively. The width $W(L,t)$ of the interface, encapsulating its fluctuations, adheres to robust scaling laws across system sizes and observation times, constituting the KPZ universality class.

Stochastic Loewner Evolution with Nonlinear Driving

SLE, originally devised for the scaling limits of 2D conformally invariant random curves, is defined by the Loewner differential equation

$$\frac{\partial g_s(z)}{\partial s} = \frac{2}{g_s(z) - U_s}, \qquad g_0(z) = z,$$

where $U_s$ is the real-valued driving function. By introducing a nonlinear, state-dependent stochastic process as $U_s$, the connection to 1D KPZ dynamics emerges. Specifically, the driving function evolves as

$$\frac{d U_s}{ds} = -\frac{x^2 + y^2}{2y} - \sqrt{\kappa}\frac{2y}{x^2 + y^2}\frac{dB_s}{ds},$$

and, via coordinate transformations (notably $h(x,t)$0), this SLE variant reproduces KPZ-like stochastic dynamics for suitably defined $h(x,t)$1.

Analytical Correspondence

By constructing the height function as

$h(x,t)$2

and mapping the SLE-driven state variable $h(x,t)$3 as per the Langevin equation with time-dependent coefficients, the statistical properties of the KPZ interface are recapitulated. Analytical work establishes that the variance $h(x,t)$4 demonstrates the correct scaling in $h(x,t)$5, consistent with KPZ exponents, under the approximation that higher-order terms ($h(x,t)$6) are negligible for small $h(x,t)$7. The ensemble dynamics of $h(x,t)$8—including mean, curvature, and fluctuations—match those derived from the KPZ equation, conditional on the identification of parameters and appropriate noise correspondences.

Loewner Entropy and Universality Classification

The Loewner entropy,

$h(x,t)$9

where $\alpha=1/2$0 is the probability density function of the SLE driving force, serves as a complexity measure of the interface dynamics. The paper demonstrates analytically and numerically that $\alpha=1/2$1 in the KPZ regime. This scaling encodes the universality class directly into the conformal stochastic process, implying that the broad class of 1D nonlinear growth processes within KPZ universality can be recapitulated and classified through the behavior of Loewner entropy.

Numerical Verification

Two principal numerical experiments are reported. First, discrete approximations of the modified SLE yield $\alpha=1/2$2 trajectories whose width $\alpha=1/2$3 exhibits the expected KPZ scalings: $\alpha=1/2$4 for short times (growth regime) and $\alpha=1/2$5 subsequently (saturation regime, mapped via system size scaling $\alpha=1/2$6). The scaling crossovers and exponent values match analytical predictions. Second, computation of the Loewner driving force using zipper algorithms and evaluation of $\alpha=1/2$7 confirms the $\alpha=1/2$8 scaling, substantiating $\alpha=1/2$9, as analytically deduced.

Discussion of Implications

This study provides a bridge between nonlinear stochastic growth (KPZ) and conformal stochastic processes (SLE), offering an alternative analytical and computational toolset for questions previously constrained by the intractability of direct KPZ solutions. The identification of Loewner entropy as a classifier for universality classes suggests future research opportunities: extending this approach to other stochastic nonlinear PDEs, probing its validity beyond $\beta=1/3$0, and examining experimental systems (e.g., pattern formation, biological morphogenesis, or interfacial phenomena in driven systems) for SLE-like fingerprints. Furthermore, the conformal invariance property of Loewner entropy hints at deeper structural connections between nonequilibrium growth, geometry, and information-theoretic measures.

Conclusion

By establishing a detailed correspondence between the 1D KPZ equation and an SLE with nonlinear driving, this work delivers both analytical and numerical results validating the equivalence at the level of stochastic dynamics, scaling laws, and complexity (via Loewner entropy). These results generalize the applicability of SLE methodologies to a fundamental universality class in nonequilibrium statistical physics and posit Loewner entropy as a diagnostic tool for characterizing stochastic nonlinear growth beyond standard frameworks.

Citation: "Description of KPZ interface growth by stochastic Loewner evolution" (2604.03711)