Kardar-Parisi-Zhang Equation Research Articles

An invariance principle for the 1D KPZ equation

An invariance principle for the 1D KPZ equation

High moments of the SHE in the clustering regimes

We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under the delta(-like) initial condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar–Parisi–Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.

KPZ equation limit of sticky Brownian motion

We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the (1+1) dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in [66,7] and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an N-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order (log⁡N)2.

Derivation of Coupled KPZ Equations from Interacting Diffusions Driven by a Single-Site Potential

Derivation of Coupled KPZ Equations from Interacting Diffusions Driven by a Single-Site Potential

Analytical and numerical study of diffusion propelled surface growth phenomena

To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

Scaling regimes of the one-dimensional phase turbulence in the deterministic complex Ginzburg-Landau equation.

We consider the one-dimensional deterministic complex Ginzburg-Landau equationin the regime of phase turbulence, where the order parameter displays a defect-free chaotic phase dynamics, which maps to the Kuramoto-Sivashinsky equation, characterized by negative viscosity and a modulational instability at linear level. In this regime, the dynamical behavior of the large wavelength modes is captured by the Kardar-Parisi-Zhang (KPZ) universality class, determining their universal scaling and their statistical properties. These modes exhibit the characteristic KPZ superdiffusive scaling with the dynamical critical exponent z=3/2. We present numerical evidence of the existence of an additional scale-invariant regime, with the dynamical exponent z=1, emerging at scales which are intermediate between the microscopic ones, intrinsic to the modulational instability, and the macroscopic ones. We argue that this new scaling regime belongs to the universality class corresponding to the inviscid limit of the KPZ equation.

Pinning-depinning transitions in two classes of discrete elastic-string models in (2+1)-dimensions

The pinning-depinning phase transitions of interfaces for two classes of discrete elastic-string models are investigated numerically. In the (1+1)-dimensions, we revisit these two elastic-string models with slight modification to the growth rule, and compare the estimated values with the previous numerical and experimental results. For the (2+1)-dimensional case, we perform extensive simulations on pinning-depinning transitions in these discrete models with quenched disorder. For full comparisons in the physically relevant spatial dimensions, we also perform numerically two distinct universality classes, including the quenched Edwards–Wilkinson, and the quenched Kardar–Parisi–Zhang equations with and without external driving forces. The critical exponents of these systems in the presence of quenched disorder are numerically estimated. Our results show that the critical exponents satisfy scaling relations well, and these two discrete elastic-string models do not fall into the existing universality classes. In order to visually comparisons of these discrete systems with quenched disorder in the (2+1)-dimensional cases, we present surface morphologies with various external driving forces during the saturated time regimes. The relationships between surface morphologies, scaling exponents and correlation length are also revealed.

Non-stationary KPZ equation from ASEP with slow bonds

Non-stationary KPZ equation from ASEP with slow bonds

Field-theoretic approach to neutron noise in nuclear reactors.

An operating nuclear reactor is designed to maintain a sustained fission chain reaction in its core, which results from a delicate balance between neutron creations (i.e., fissions) and total absorptions. This balance is associated with random fluctuations that can have two, very different, origins. A distinction must thus be made between low-power noise, whose origin lies in the inherently stochastic nature of neutron interactions with matter, and high-power noise, whose origin lies in the particular thermomechanical constraints associated with the environment in which neutrons propagate. Modeling the behavior of this noisy neutron population with the help of stochastic differential equations, we first show how the Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) formalism, providing a field theoretical representation of the problem, reveals a convenient and adapted tool for the calculation of observable consequences of neutron noise. In particular, we show how the MSRJD approach is capable of encompassing both types of neutron noises in the same formalism. Emphasizing then on power noise, it is shown how the self-sustained chain reaction developing in a reactor core might be sensitive to noise-induced transitions. Establishing an unprecedented connection between the neutron population evolving in a reactor core and the celebrated Kardar-Parisi-Zhang (KPZ) equation, we indeed find evidence that a noisy reactor core power distribution might be subject to a process analogous to the roughening transition, well-known to occur in systems described by the KPZ equation.

Extensive numerical simulations on competitive growth between the Edwards–Wilkinson and Kardar–Parisi–Zhang universality classes

Extensive numerical simulations and scaling analysis are performed to investigate competitive growth between the linear and nonlinear stochastic dynamic growth systems, which belong to the Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) universality classes, respectively. The linear growth systems include the EW equation and the model of random deposition with surface relaxation (RDSR), the nonlinear growth systems involve the KPZ equation and typical discrete models including ballistic deposition (BD), etching, and restricted solid on solid (RSOS). The scaling exponents are obtained in both the (1 + 1)- and (2 + 1)-dimensional competitive growth with the nonlinear growth probability p and the linear proportion 1 – p. Our results show that, when p changes from 0 to 1, there exist non-trivial crossover effects from EW to KPZ universality classes based on different competitive growth rules. Furthermore, the growth rate and the porosity are also estimated within various linear and nonlinear growths of cooperation and competition.

Logarithmic or algebraic: Roughening of an active Kardar-Parisi-Zhang surface.

The Kardar-Parisi-Zhang (KPZ) equationsets the universality class for growing and roughening of nonequilibrium surfaces without any conservation law and nonlocal effects. We argue here that the KPZ equationcan be generalized by including a symmetry-permitted nonlocal nonlinear term of active origin that is of the same order as the one included in the KPZ equation. Including this term, the 2D active KPZ equationis stable in some parameter regimes, in which the interface conformation fluctuations exhibit sublogarithmic or superlogarithmic roughness, with nonuniversal exponents, giving positional generalized quasi-long-ranged order. For other parameter choices, the model is unstable, suggesting a perturbatively inaccessible algebraically rough interface or positional short-ranged order. Our model should serve as a paradigmatic nonlocal growth equation.

Boundary current fluctuations for the half‐space ASEP and six‐vertex model

AbstractWe study fluctuations of the current at the boundary for the half‐space asymmetric simple exclusion process (ASEP) and the height function of the half‐space six‐vertex model at the boundary at large times. We establish a phase transition depending on the effective density of particles at the boundary, with Gaussian symplectic ensemble (GSE) and Gaussian orthogonal ensemble (GOE) limits as well as the Baik–Rains crossover distribution near the critical point. This was previously known for half‐space last‐passage percolation, and recently established for the half‐space log‐gamma polymer and Kardar–Parisi–Zhang equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half‐space six‐vertex model and a half‐space Hall–Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half‐space ASEP and six‐vertex model, indicating a hidden free fermionic structure.

Space–time fluctuation of the Kardar–Parisi–Zhang equation in d≥3 and the Gaussian free field

Space–time fluctuation of the Kardar–Parisi–Zhang equation in d≥3 and the Gaussian free field

Mesoscopic Averaging of the Two-Dimensional KPZ Equation

Mesoscopic Averaging of the Two-Dimensional KPZ Equation

Unpredicted Scaling of the One-Dimensional Kardar-Parisi-Zhang Equation.

The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [C. Cartes et al., The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. R. Soc. A 380, 20210090 (2022).PTRMAD1364-503X10.1098/rsta.2021.0090; E. Rodríguez-Fernández et al., Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022).PRESCM2470-004510.1103/PhysRevE.106.024802] unveiled a new scaling, with a critical dynamical exponent z=1 different from the KPZ one z=3/2. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite nonlinear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields z=1. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wave numbers. Both scaling functions accurately match the one from the numerical simulations.

Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations.

The out-of-time-ordered correlator (OTOC) has emerged as an interesting object in both classical and quantum systems for probing the spatial spread and temporal growth of initially local perturbations in spatially extended chaotic systems. Here, we study the (classical) OTOC and its "light cone" in the nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical simulations. We also show that the linearized KS equationexhibits a qualitatively similar OTOC and light cone, which can be understood via a saddle-point analysis of the linearly unstable modes. Given the deep connection between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ, which is stochastic) equations, we also explore the OTOC in the KPZ equation. While our numerical results in the KS case are expected to hold in the continuum limit, for the KPZ case it is valid in a discretized version of the KPZ equation. More broadly, our work unravels the intrinsic interplay between noise/instability, nonlinearity, and dissipation in partial differential equations(deterministic or stochastic) through the lens of OTOC.

Stationary measure for the open KPZ equation

AbstractWe provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0,1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters u and v, respectively. When , we uniquely characterize the constructed stationary measures through their multipoint Laplace transform, which we prove is given in terms of a stochastic process that we call the continuous dual Hahn process. Our work relies on asymptotic analysis of Bryc and Wesołowski's Askey–Wilson process formulas for the open ASEP stationary measure (which in turn arise from Uchiyama, Sasamoto and Wadati's Askey‐Wilson Jacobi matrix representation of Derrida et al.'s matrix product ansatz) in conjunction with Corwin and Shen's proof that open ASEP converges to open KPZ under weakly asymmetric scaling.

Directed mean curvature flow in noisy environment

AbstractWe consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite‐dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

Effect of Random Environment on Kinetic Roughening: Kardar–Parisi–Zhang Model with a Static Noise Coupled to the Navier–Stokes Equation

Effect of Random Environment on Kinetic Roughening: Kardar–Parisi–Zhang Model with a Static Noise Coupled to the Navier–Stokes Equation

Stationary measures for the log-gamma polymer and KPZ equation in half-space

Stationary measures for the log-gamma polymer and KPZ equation in half-space