The one-dimensional KPZ equation and its universality class

Abstract

Our understanding of the one-dimensional KPZ equation, \textit{alias} noisy Burgers equation, has advanced substantially over the past five years. We provide a non-technical review, where we limit ourselves to the stochastic PDE and lattice type models approximating it.

24 pages, 2 figures

References in corpus (47)

• Level-Spacing Distributions and the Airy Kernel
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• A theory of regularity structures
• Free-energy distribution of the directed polymer at high temperature
• Level-Spacing Distributions and the Airy Kernel
• Growing interfaces uncover universal fluctuations behind scale invariance
• Asymptotics in ASEP with Step Initial Condition
• On the Numerical Evaluation of Fredholm Determinants
• Fluctuation properties of the TASEP with periodic initial configuration
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• An exact solution for the KPZ equation with flat initial conditions
• Scaling for a one-dimensional directed polymer with boundary conditions
• Random matrices and determinantal processes
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• Spatial correlations of the 1D KPZ surface on a flat substrate
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Exact solution for the stationary Kardar-Parisi-Zhang equation
• The intermediate disorder regime for directed polymers in dimension $1+1$
• Integral Formulas for the Asymmetric Simple Exclusion Process
• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
• The KPZ equation with flat initial condition and the directed polymer with one free end
• From duality to determinants for q-TASEP and ASEP
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Stationary correlations for the 1D KPZ equation
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• A determinantal formula for the GOE Tracy-Widom distribution
• Scale Invariance of the PNG Droplet and the Airy Process
• Universality of slow decorrelation in KPZ growth
• Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers
• The One-dimensional KPZ Equation and the Airy Process
• Finite time corrections in KPZ growth models
• Universal fluctuations in radial growth models belonging to the KPZ universality class
• Dynamics of the attractive 1D Bose gas: analytical treatment from integrability
• Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions
• Statistics of circular interface fluctuations in an off-lattice Eden model
• On Asymptotics for the Airy Process
• The height distribution of the KPZ equation with sharp wedge initial condition: numerical evaluations
• Two-point free energy distribution function in (1+1) directed polymers
• A Pfaffian representation for flat ASEP
• Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
• Scaling limit for Brownian motions with one-sided collisions
• Two-point generating function of the free energy for a directed polymer in a random medium
• On the equal time two-point distribution of the one-dimensional KPZ equation by replica
• The propagator of the attractive delta-Bose gas in one dimension
• Height fluctuations for the stationary KPZ equation
• Renormalization Group and Stochastic PDE's
• KPZ equation, its renormalization and invariant measures

Cited by in corpus (125)

• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• KPZ reloaded
• Fibonacci family of dynamical universality classes
• Universal spin dynamics in infinite-temperature one-dimensional quantum magnets
• Superdiffusion from emergent classical solitons in quantum spin chains
• Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation
• Anomalous relaxation and the high-temperature structure factor of XXZ spin chains
• Dynamical phase transition in large deviation statistics of the Kardar-Parisi-Zhang equation
• Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $\lambda H<0$
• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Short-time height distribution in 1d KPZ equation: starting from a parabola
• Finite-time fluctuations for the totally asymmetric exclusion process
• Universal scaling in active single-file dynamics
• On Time Correlations for KPZ Growth in One Dimension
• Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface
• Shift-invariance for vertex models and polymers
• Universal KPZ scaling in noisy hybrid quantum circuits
• The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions
• Dynamical criticality of magnetization transfer in integrable spin chains
• Prethermalization and Thermalization of a Quenched Interacting Luttinger Liquid
• Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process
• Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions
• Time-time covariance for last passage percolation with generic initial profile
• Anomalous Dynamics and Equilibration in the Classical Heisenberg Chain
• Reflected Brownian motions in the KPZ universality class
• 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
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• 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
• Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
• Kardar-Parisi-Zhang Universality of the Nagel-Schreckenberg Model
• Short time large deviations of the KPZ equation
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• How flat is flat in random interface growth?
• Geometry of geodesics through Busemann measures in directed last-passage percolation
• On global solutions of the random Hamilton-Jacobi equations and the KPZ problem
• Energy solutions of KPZ are unique
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• 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
• Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media
• Stationary half-space last passage percolation
• Hydrodynamic Equations for the Toda Lattice
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• The 1+1 dimensional Kardar-Parisi-Zhang equation: more surprises
• Extreme boundary conditions and random tilings
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• Riemann surfaces for KPZ with periodic boundaries
• 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
• Probability distributions for directed polymers in random media with correlated noise
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• First Passage Time for Many Particle Diffusion in Space-Time Random Environments
• Extrapolation methods and Bethe ansatz for the asymmetric exclusion process
• The lower tail of the half-space KPZ equation
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Local average height distribution of fluctuating interfaces
• Hall-Littlewood-PushTASEP and its KPZ limit
• A (2+1)-dimensional Anisotropic KPZ growth model with a smooth phase
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• Population Extinction on a Random Fitness Seascape
• KPZ equation with a small noise, deep upper tail and limit shape
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Riemann surface for TASEP with periodic boundaries
• KPZ equation limit of sticky Brownian motion
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Extreme statistics of non-intersecting Brownian paths
• Lectures on singular stochastic PDEs
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Anomalous current fluctuations from Euler hydrodynamics
• Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence
• Integrability in the weak noise theory
• Fluctuation Exponents of the KPZ equation on a large torus
• Midpoint distribution of directed polymers in the stationary regime: exact result through linear response
• Effects of a local defect on one-dimensional nonlinear surface growth
• Universality of deterministic KPZ
• Efimov effect at the Kardar-Parisi-Zhang roughening transition
• High temperature behaviors of the directed polymer on a cylinder
• Thermal transport of confined water molecules in quasi-one-dimensional nanotubes
• KPZ fluctuations in finite volume
• Weak universality of dynamical ${\Phi }_3^4$: non-Gaussian noise
• On the exponent governing the correlation decay of the Airy${}_1$ process
• Uniqueness and ergodicity of stationary directed polymers on ${\mathbb{Z}}^2$
• Local KPZ behavior under arbitrary scaling limits
• The KPZ equation and moments of random matrices
• Large deviation bounds for the Airy point process
• A PDE hierarchy for directed polymers in random environments
• From the Riemann surface of TASEP to ASEP
• The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency
• Comments on David Aldous and Persi Diaconis' "Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem"
• Brownian regularity for the KPZ line ensemble
• Analytical Methods and Field Theory for Disordered Systems
• Localization length of the $1+1$ continuum directed random polymer
• Monitored Fluctuating Hydrodynamics
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• Gumbel laws in the symmetric exclusion process
• Multiplicative SHE limit of random walks in space-time random environments
• KPZ on torus: Gaussian fluctuations
• Short- and long-time path tightness of the continuum directed random polymer
• Superdiffusion transition for a phonon Boltzmann equation
• Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth
• A Seascape Origin of Richards Growth
• Parameter Permutation Symmetry in Particle Systems and Random Polymers
• Random attraction in the TASEP model
• Characterization of $H$-Brownian Gibbsian line ensembles
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Gaussian fluctuations from the 2D KPZ equation
• Infection fronts in randomly varying transmission-rate media
• Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation
• Universal properties of the Kardar-Parisi-Zhang equation with quenched columnar disorders
• High moments of the SHE in the clustering regimes
• Blocks and Gaps in the Asymmetric Simple Exclusion Process: Asymptotics
• On the $q$-TASEP with a random initial condition
• Microscopic interplay of temperature and disorder of a one-dimensional elastic interface
• A Microscopic Derivation Of Coupled Spde's With A Kpz Flavor
• Full Statistics of Regularized Local Energy Density in a Freely Expanding Kipnis-Marchioro-Presutti Gas
• The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalised to Kardar-Parisi-Zhang scaling
• Non-existence of bi-infinite polymer Gibbs measures
• The half-space Airy stat process