An appetizer to modern developments on the Kardar-Parisi-Zhang universality class

Abstract

The Kardar-Parisi-Zhang (KPZ) universality class describes a broad range of non-equilibrium fluctuations, including those of growing interfaces, directed polymers and particle transport, to name but a few. Since the year 2000, our understanding of the one-dimensional KPZ class has been completely renewed by mathematical physics approaches based on exact solutions. Mathematical physics has played a central role since then, leading to a myriad of new developments, but their implications are clearly not limited to mathematics -- as a matter of fact, it can also be studied experimentally. The aim of these lecture notes is to provide an introduction to the field that is accessible to non-specialists, reviewing basic properties of the KPZ class and highlighting main physical outcomes of mathematical developments since the year 2000. It is written in a brief and self-contained manner, with emphasis put on physical intuitions and implications, while only a small (and mostly not the latest) fraction of mathematical developments could be covered. Liquid-crystal experiments by the author and coworkers are also reviewed.

34 pages, 16 figures, 1 table; lecture notes prepared for Summer School "Fundamental Problems in Statistical Physics XIV" (16-29 July 2017, Bruneck, Italy); descriptions on growth under quenched noise are significantly updated in v2; otherwise most changes are stylistic and made for better accessibility to non-specialists

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• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Growing interfaces uncover universal fluctuations behind scale invariance
• Asymptotics in ASEP with Step Initial Condition
• Exact scaling functions for one-dimensional stationary KPZ growth
• Fluctuation properties of the TASEP with periodic initial configuration
• An exact solution for the KPZ equation with flat initial conditions
• The one-dimensional KPZ equation and its universality class
• The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
• The KPZ fixed point
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Dynamical structure factor of one-dimensional Bose gases: experimental signatures of beyond-Luttinger liquid physics
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Universal correlators and distributions as experimental signatures of 2+1 Kardar-Parisi-Zhang growth
• Numerical estimate of the Kardar Parisi Zhang universality class in (2 + 1) dimensions
• Kardar-Parisi-Zhang universality class in 2+1 dimensions: Universal geometry-dependent distributions and finite-time corrections
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Slow decorrelations in KPZ growth
• Memory and universality in interface growth
• Temperature effect on (2+1) experimental Kardar-Parisi-Zhang growth
• Unbinding transition in semi-infinite two-dimensional localized systems
• Dynamical universality classes of simple growth and lattice gas models
• Intrinsic geometry approach to surface kinetic roughening
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Universality of critically pinned interfaces in 2-dimensional isotropic random media

Cited by in corpus (111)

• Observation of KPZ universal scaling in a one-dimensional polariton condensate
• Anomalous spin diffusion in one-dimensional antiferromagnets
• The directed landscape
• Superuniversality of superdiffusion
• Universality classes of spin transport in one-dimensional isotropic magnets: the onset of logarithmic anomalies
• Universality in driven open quantum matter
• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Quantum Criticality in Open Quantum Spin Chains with Nonreciprocity
• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics
• Integrable Matrix Models in Discrete Space-Time
• 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
• Dynamical criticality of magnetization transfer in integrable spin chains
• Universal anomalous fluctuations in charged single-file systems
• Time-time covariance for last passage percolation with generic initial profile
• Functional renormalisation group for turbulence
• Strong coupling in conserved surface roughening: A new universality class?
• Uniform convergence to the Airy line ensemble
• Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions
• Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface
• Universal phenomenology at critical exceptional points of nonequilibrium $O(N)$ models
• Half-space stationary Kardar-Parisi-Zhang equation
• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• Impact of Dissipation on Universal Fluctuation Dynamics in Open Quantum Systems
• Field-Theoretic Thermodynamic Uncertainty Relation -- General formulation exemplified with the Kardar-Parisi-Zhang equation
• Kardar-Parisi-Zhang Equation with temporally correlated noise: a non-perturbative renormalization group approach
• 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
• The 1+1 dimensional Kardar-Parisi-Zhang equation: more surprises
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Kardar-Parisi-Zhang universality in discrete two-dimensional driven-dissipative exciton polariton condensates
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• Riemann surfaces for KPZ with periodic boundaries
• Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation
• Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces
• Emergent Kardar-Parisi-Zhang phase in quadratically driven condensates
• Kardar-Parisi-Zhang Physics in the Density Fluctuations of Localized Two-Dimensional Wave Packets
• Spatiotemporal scaling of two-dimensional nonequilibrium exciton-polariton systems with weak interactions
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices
• Kardar-Parisi-Zhang universality class for the critical dynamics of reaction-diffusion fronts
• Circular Kardar-Parisi-Zhang interfaces evolving out of the plane
• KPZ statistics of second class particles in ASEP via mixing
• 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
• Spreading fronts of wetting liquid droplets: microscopic simulations and universal fluctuations
• Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\ge 3$ and the Gaussian free field
• The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class
• Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
• Transition between chaotic and stochastic universality classes of kinetic roughening
• The unpredicted scaling of the one-dimensional Kardar-Parisi-Zhang equation
• Phase diagram of one-dimensional driven-dissipative exciton-polariton condensates
• Anomalous current fluctuations from Euler hydrodynamics
• Stirred Kardar-Parisi-Zhang equation with quenched random noise: Emergence of induced nonlinearity
• Dynamically accelerated cover times
• Effects of turbulent environment on the surface roughening: The Kardar-Parisi-Zhang model coupled to the stochastic Navier-Stokes equation
• Domain wall fluctuations of the six-vertex model at the ice point
• Kardar-Parisi-Zhang universality class in the synchronization of oscillator lattices with time-dependent noise
• Exactly solvable models of growing interfaces and lattice gases: the Arcetri models, ageing and logarithmic sub-ageing
• Kardar-Parisi-Zhang fluctuations in the synchronization dynamics of limit-cycle oscillators
• Efimov effect at the Kardar-Parisi-Zhang roughening transition
• KPZ fluctuations in finite volume
• Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• Kardar-Parisi-Zhang scaling in time-crystalline matter
• On the exponent governing the correlation decay of the Airy${}_1$ process
• Scaling regimes of the one-dimensional phase turbulence in the deterministic complex Ginzburg-Landau equation
• Analytic iteration procedure for solitons and traveling wavefronts with sources
• A Unified Interface Model for Dissipative Transport of Bosons and Fermions
• Large-scale kinetic roughening behavior of coffee-ring fronts
• Beyond-mean-field corrections to the blueshift of a driven-dissipative exciton-polariton condensate
• Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations
• Width scaling of an interface constrained by a membrane
• BLUES iteration applied to nonlinear ordinary differential equations for wave propagation and heat transfer
• Microscopic fluctuations in the spreading fronts of circular wetting liquid droplets
• Influence of coupling symmetries and noise on the critical dynamics of synchronizing oscillator lattices
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• How to exploit driving and dissipation to stabilize and manipulate quantum many-body states
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• Fluctuations of stochastic charged cellular automata
• Phase diagram and universal scaling regimes of two-dimensional exciton-polariton Bose-Einstein condensates
• Universality in many-body driven systems with an umbilic point
• Approach to stationarity for the KPZ fixed point with boundaries
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth
• Hydrodynamic fluctuations of stochastic charged cellular automata
• From Kardar-Parisi-Zhang scaling to soliton proliferation in Josephson junction arrays
• Universal interface fluctuations in the contact process
• Universal properties of the Kardar-Parisi-Zhang equation with quenched columnar disorders
• Infection fronts in randomly varying transmission-rate media
• The non-perturbative sides of the Kardar-Parisi-Zhang equation
• The Two Scaling Regimes of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
• Mechanism underlying dynamic scaling properties observed in the contour of spreading epithelial monolayer
• Crossover from BKT-Rough to KPZ-Rough Surfaces for Interface-Limited Crystal Growth/Recession
• Nonconserved critical dynamics of the two-dimensional Ising model as a surface kinetic roughening process
• Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments
• A Simple XY Model for Cascade Transfer
• Microscopic interplay of temperature and disorder of a one-dimensional elastic interface
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• Full Statistics of Regularized Local Energy Density in a Freely Expanding Kipnis-Marchioro-Presutti Gas
• Universal fluctuations of localized two interacting particles in one dimension
• Current fluctuations for the second class particle : joint statistics
• Divergent stiffness of one-dimensional growing interfaces
• 2D Anderson Localization and KPZ sub-Universality Classes : sensitivity to boundary conditions and insensitivity to symmetry classes
• Unveiling the different scaling regimes of the one-dimensional Kardar-Parisi-Zhang--Burgers equation using the functional renormalisation group
• The half-space Airy stat process
• Shape-specific fluctuations of an active colloidal interface
• Exponents and front fluctuations in the quenched Kardar-Parisi-Zhang universality class of one- and two- dimensional interfaces
• Intrinsic Step Jamming in Nanometer-Scale KPZ-like Rough Surfaces under Interface-Limited Crystal Growth and Retreat