A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces

Abstract

The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we i) pay debts to heroic predecessors, ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of $d$=$\infty$ KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.

Mini-review, mixed w/ new results. 18 pages, 6 figures. Journal of Statistical Physics, in press; v2: corrects typos, adds refs

References in corpus (21)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Growing interfaces uncover universal fluctuations behind scale invariance
• Asymptotics in ASEP with Step Initial Condition
• An exact solution for the KPZ equation with flat initial conditions
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Universal correlators and distributions as experimental signatures of 2+1 Kardar-Parisi-Zhang growth
• Kardar-Parisi-Zhang universality class in 2+1 dimensions: Universal geometry-dependent distributions and finite-time corrections
• Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension
• Statistics of circular interface fluctuations in an off-lattice Eden model
• Multi-surface coding simulations of the restricted solid-on-solid model in four dimensions
• Interface fluctuations for deposition on enlarging flat substrates
• Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder
• Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces
• From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation
• Spherical model of growing interfaces
• Random geometry and the Kardar-Parisi-Zhang universality class
• Numerical study of the directed polymer in a 1+3 dimensional random medium
• Universal width distributions in non-Markovian Gaussian processes
• Intrinsic geometry approach to surface kinetic roughening
• KPZ universality class and the anchored Toom interface

Cited by in corpus (151)

• Operator Spreading in Random Unitary Circuits
• Quantum Entanglement Growth Under Random Unitary Dynamics
• Random Quantum Circuits
• Emergent statistical mechanics of entanglement in random unitary circuits
• Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet
• Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion
• Extreme value statistics of correlated random variables: a pedagogical review
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems
• Velocity-dependent Lyapunov exponents in many-body quantum, semi-classical, and classical chaos
• Fibonacci family of dynamical universality classes
• Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation
• Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature
• Dynamical phase transition in large deviation statistics of the Kardar-Parisi-Zhang equation
• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $\lambda H<0$
• Exact short-time height distribution in 1D KPZ equation with Brownian initial condition
• Coulomb-gas electrostatics controls large fluctuations of the KPZ equation
• Short-time height distribution in 1d KPZ equation: starting from a parabola
• Finite-time fluctuations for the totally asymmetric exclusion process
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics
• Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface
• Non-linear fluctuating hydrodynamics for KPZ scaling in isotropic spin chains
• Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation
• Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions
• Simple derivation of the $(-\lambda H{)}^{5/2}$ tail for the 1D KPZ equation
• Initial pseudo-steady state & asymptotic KPZ universality in semiconductor on polymer deposition
• Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Facilitated exclusion process
• Searching for the Tracy-Widom distribution in nonequilibrium processes
• Large fluctuations of the KPZ equation in a half-space
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Kardar-Parisi-Zhang Interfaces with Inward Growth
• Half-space stationary Kardar-Parisi-Zhang equation
• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Exact scaling solution of the mode coupling equations for non-linear fluctuating hydrodynamics in one dimension
• Kardar-Parisi-Zhang Universality of the Nagel-Schreckenberg Model
• Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• From Kardar-Parisi-Zhang scaling to explosive desynchronization in arrays of limit-cycle oscillators
• The Kardar-Parisi-Zhang exponents for the $2+1$ dimensions
• Geometry of geodesics through Busemann measures in directed last-passage percolation
• Field-Theoretic Thermodynamic Uncertainty Relation -- General formulation exemplified with the Kardar-Parisi-Zhang equation
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• Topology and the Kardar-Parisi-Zhang universality class
• A Riemann-Hilbert approach to the lower tail of the KPZ equation
• Non-universality of front fluctuations for compact colonies of non-motile bacteria
• Tail of the two-time height distribution for KPZ growth in one dimension
• Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• The 1+1 dimensional Kardar-Parisi-Zhang equation: more surprises
• Eden model with nonlocal growth rules and the kinetic roughening in biological systems
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class
• Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation
• Riemann surfaces for KPZ with periodic boundaries
• Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates
• Geometry dependence in linear interface growth
• Bacterial range expansions on a growing front: Roughness, Fixation, and Directed Percolation
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Entanglement entropy growth in stochastic conformal field theory and the KPZ class
• Nonequilibrium Steady State of a Weakly-Driven Kardar-Parisi-Zhang Equation
• Driven interfaces: from flow to creep through model reduction
• Probability distributions for directed polymers in random media with correlated noise
• Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Liouville field theory and log-correlated Random Energy Models
• Kardar-Parisi-Zhang universality class for the critical dynamics of reaction-diffusion fronts
• Charge-current correlation equalities for quantum systems far from equilibrium
• Kardar-Parisi-Zhang universality class in ($d+1$)-dimensions
• Extrapolation methods and Bethe ansatz for the asymmetric exclusion process
• Generating stochastic trajectories with global dynamical constraints
• $1/f^{\alpha }$ power spectrum in the Kardar-Parisi-Zhang universality class
• Spreading fronts of wetting liquid droplets: microscopic simulations and universal fluctuations
• Dynamical universality of the contact process
• Local average height distribution of fluctuating interfaces
• A (2+1)-dimensional growth process with explicit stationary measures
• Hallmarks of the Kardar-Parisi-Zhang universality class elicited by scanning probe microscopy
• Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ
• Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica
• Scaling of local roughness distributions
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Transition between chaotic and stochastic universality classes of kinetic roughening
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• The unpredicted scaling of the one-dimensional Kardar-Parisi-Zhang equation
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Gaussian statistics as an emergent symmetry of the stochastic Burgers equation
• Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice
• Kardar-Parisi-Zhang Universality in First-Passage Percolation: the Role of Geodesic Degeneracy
• Width and extremal height distributions of fluctuating interfaces with window boundary conditions
• A Schelling model with a variable treshold in a closed city segregation model. Analysis of the universality classes
• A lattice gas model for generic one-dimensional Hamiltonian Systems
• Mutually avoiding paths in random media and largests eigenvalues of random matrices
• KPZ fluctuations in finite volume
• Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• KPZ modes in $d$-dimensional directed polymers
• Non-local meta-conformal invariance in diffusion-limited erosion
• First-Passage Percolation under extreme disorder: from bond-percolation to Kardar-Parisi-Zhang universality
• Uniqueness and ergodicity of stationary directed polymers on ${\mathbb{Z}}^2$
• Critical dynamics of the jamming transition in one-dimensional nonequilibrium lattice-gas models
• Power countings versus physical scalings in disordered elastic systems - Case study of the one-dimensional interface
• Stochastic Dynamics of Growing Young Diagrams and Their Limit Shapes
• Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations
• Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time
• Microscopic fluctuations in the spreading fronts of circular wetting liquid droplets
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• Inviscid limit of the active interface equations
• Phase diagram and universal scaling regimes of two-dimensional exciton-polariton Bose-Einstein condensates
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Analytical Methods and Field Theory for Disordered Systems
• Hyperskewness of $(1+1)$-dimensional KPZ Height Fluctuations
• Rough or crumpled: Strong coupling phases of a generalized Kardar-Parisi-Zhang surface
• Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth
• One-point height fluctuations and two-point correlators of $(2+1)$ cylindrical KPZ systems
• The role of the non-linearity in controlling the surface roughness in the one-dimensional Kardar--Parisi--Zhang growth process
• Tracking evaporative cooling of a mesoscopic atomic quantum gas in real time
• Quest for the golden ratio universality class
• Parameter Permutation Symmetry in Particle Systems and Random Polymers
• Quasi-stationary states in nonlocal stochastic growth models with infinitely many absorbing states
• Dimensional crossover in Kardar-Parisi-Zhang growth
• Generic phase coexistence in the totally asymmetric kinetic Ising model
• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
• Infection fronts in randomly varying transmission-rate media
• Universal fluctuations of global geometrical measurements in planar clusters
• Weak pinning and long-range anticorrelated motion of phase boundaries in driven diffusive systems
• The Two Scaling Regimes of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• Universal interface fluctuations in the contact process
• Height distributions in interface growth: The role of the averaging process
• Universal properties of the Kardar-Parisi-Zhang equation with quenched columnar disorders
• Steady-state skewness and kurtosis from renormalized cumulants in $(2+1)$-dimensional stochastic surface growth
• The non-perturbative sides of the Kardar-Parisi-Zhang equation
• Nonconserved critical dynamics of the two-dimensional Ising model as a surface kinetic roughening process
• The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalised to Kardar-Parisi-Zhang scaling
• Universal fluctuations of localized two interacting particles in one dimension
• Rate of convergence for polymers in a weak disorder
• 2D Anderson Localization and KPZ sub-Universality Classes : sensitivity to boundary conditions and insensitivity to symmetry classes
• Exponents and front fluctuations in the quenched Kardar-Parisi-Zhang universality class of one- and two- dimensional interfaces
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization
• Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime
• Microscopic interplay of temperature and disorder of a one-dimensional elastic interface
• Roughening of $k$-mer growing interfaces in stationary regimes
• Divergent stiffness of one-dimensional growing interfaces