A pedestrian's view on interacting particle systems, KPZ universality, and random matrices

Abstract

These notes are based on lectures delivered by the authors at a Langeoog seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a mixed audience of mathematicians and theoretical physicists. After a brief outline of the basic physical concepts of equilibrium and nonequilibrium states, the one-dimensional simple exclusion process is introduced as a paradigmatic nonequilibrium interacting particle system. The stationary measure on the ring is derived and the idea of the hydrodynamic limit is sketched. We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and explain the associated universality conjecture for surface fluctuations in growth models. This is followed by a detailed exposition of a seminal paper of Johansson that relates the current fluctuations of the totally asymmetric simple exclusion process (TASEP) to the Tracy-Widom distribution of random matrix theory. The implications of this result are discussed within the framework of the KPZ conjecture.

52 pages, 4 figures; to appear in J. Phys. A: Math. Theor

References in corpus (30)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
• The one-dimensional KPZ equation: an exact solution and its universality
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Free-energy distribution of the directed polymer at high temperature
• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• Stochastic interacting particle systems out of equilibrium
• The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
• Integral Formulas for the Asymmetric Simple Exclusion Process
• A Fredholm Determinant Representation in ASEP
• Orthogonal polynomial ensembles in probability theory
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Transition between Airy_1 and Airy_2 processes and TASEP fluctuations
• The crossover regime for the weakly asymmetric simple exclusion process
• Current Distribution and random matrix ensembles for an integrable asymmetric fragmentation process
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• Second class particles and cube root asymptotics for Hammersley's process
• Limit process of stationary TASEP near the characteristic line
• Slow decorrelations in KPZ growth
• Total Current Fluctuations in ASEP
• On ASEP with Step Bernoulli Initial Condition
• Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation
• The Airy_1 process is not the limit of the largest eigenvalue in GOE matrix diffusion
• $t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb{Z}$
• Limit processes for TASEP with shocks and rarefaction fans
• Formulas for ASEP with Two-Sided Bernoulli Initial Condition
• On the Distribution of a Second Class Particle in the Asymmetric Simple Exclusion Process

Cited by in corpus (165)

• Operator Spreading in Random Unitary Circuits
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• Random Quantum Circuits
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• Emergent statistical mechanics of entanglement in random unitary circuits
• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Effects of Particle Shape on Growth Dynamics at Edges of Evaporating Colloidal Drops
• Exact Current Statistics of the ASEP with Open Boundaries
• The Exclusion Process: A paradigm for non-equilibrium behaviour
• Master equations and the theory of stochastic path integrals
• Extremely large scale simulation of a Kardar-Parisi-Zhang model using graphics cards
• The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• From GPE to KPZ: finite temperature dynamical structure factor of the 1D Bose gas
• Kardar-Parisi-Zhang universality class in 2+1 dimensions: Universal geometry-dependent distributions and finite-time corrections
• Crossover from Growing to Stationary Interfaces in the Kardar-Parisi-Zhang Class
• Evolutionary games of condensates in coupled birth-death processes
• Exact extremal statistics in the classical $1d$ Coulomb gas
• Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension
• Universal aspects of curved, flat & stationary-state Kardar-Parisi-Zhang statistics
• Some Exact Results for the Exclusion Process
• Universal fluctuations in radial growth models belonging to the KPZ universality class
• Noisy Coupled Qubits: Operator Spreading and the Fredrickson-Andersen Model
• From interacting particle systems to random matrices
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Finite-time fluctuations for the totally asymmetric exclusion process
• Phenomenology of ageing in the Kardar-Parisi-Zhang equation
• Exact description of quantum stochastic models as quantum resistors
• Statistics of circular interface fluctuations in an off-lattice Eden model
• Generic transport formula for a system driven by Markovian reservoirs
• Quantum Criticality in Open Quantum Spin Chains with Nonreciprocity
• Experimental approaches to universal out-of-equilibrium scaling laws: turbulent liquid crystal and other developments
• Universal fluctuations in KPZ growth on one-dimensional flat substrates
• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• Memory and universality in interface growth
• Current moments of 1D ASEP by duality
• Matrix Ansatz for the Fluctuations of the Current in the ASEP with Open Boundaries
• Right tail expansion of Tracy-Widom beta laws
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Non-universal parameters, corrections and universality in Kardar-Parisi-Zhang growth
• Strong coupling in conserved surface roughening: A new universality class?
• Free Fermions with a Localized Source
• Anomalous Dynamics and Equilibration in the Classical Heisenberg Chain
• Open two-species exclusion processes with integrable boundaries
• Replica approach to the KPZ equation with half Brownian motion initial condition
• Clonal interference and Muller's ratchet in spatial habitats
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Kardar-Parisi-Zhang Interfaces with Inward Growth
• Kinetics of mobile impurities and correlation functions in one-dimensional superfluids at finite temperature
• Entanglement in low-energy states of the random-hopping model
• Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation
• The bosonic skin effect: boundary condensation in asymmetric transport
• Impact of Dissipation on Universal Fluctuation Dynamics in Open Quantum Systems
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• Spherical model of growing interfaces
• Revisiting Directed Polymers with heavy-tailed disorder
• On logarithmic extensions of local scale-invariance
• Accessing Kardar-Parisi-Zhang universality sub-classes with exciton polaritons
• Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation
• The Heisenberg XX spin chain and low-energy QCD
• When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class
• Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates
• Riemann surfaces for KPZ with periodic boundaries
• Entanglement entropy growth in stochastic conformal field theory and the KPZ class
• Kardar-Parisi-Zhang scaling in the Hubbard model
• The circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues
• Scaling, cumulant ratios and height distribution of the ballistic deposition in 3+1 and 4+1 dimensions
• Smoothly-varying hopping rates in driven flow with exclusion
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Large deviations of the length of the longest increasing subsequence of random permutations and random walks
• Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices
• Competing Universalities in Kardar-Parisi-Zhang (KPZ) Growth Models
• Kardar-Parisi-Zhang universality class for the critical dynamics of reaction-diffusion fronts
• Kardar-Parisi-Zhang universality class in ($d+1$)-dimensions
• $1/f^{\alpha }$ power spectrum in the Kardar-Parisi-Zhang universality class
• Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations
• Generating stochastic trajectories with global dynamical constraints
• Dimensional fragility of the Kardar-Parisi-Zhang universality class
• Extrapolation methods and Bethe ansatz for the asymmetric exclusion process
• Random matrices and the New York City subway system
• Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation: numerical study
• Eden clusters in three-dimensions and the Kardar-Parisi-Zhang universality class
• Spreading fronts of wetting liquid droplets: microscopic simulations and universal fluctuations
• Growth Inside a Corner: The Limiting Interface Shape
• 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
• Transition between chaotic and stochastic universality classes of kinetic roughening
• Characteristic Sign Renewals of Kardar-Parisi-Zhang Fluctuations
• Melting of an Ising Quadrant
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Riemann surface for TASEP with periodic boundaries
• Kardar-Parisi-Zhang Universality in First-Passage Percolation: the Role of Geodesic Degeneracy
• A Deterministic Model for One-Dimensional Excluded Flow with Local Interactions
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice
• Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process
• Logarithmic correlators or responses in non-relativistic analogues of conformal invariance
• On the asymmetric zero-range in the rarefaction fan
• Unbalanced Langmuir kinetics affects TASEP dynamical transitions: mean-field theory
• Width and extremal height distributions of fluctuating interfaces with window boundary conditions
• Height distributions in competitive one-dimensional Kardar-Parisi-Zhang systems
• Statistics of layered zigzags: a two-dimensional generalization of TASEP
• How many longest increasing subsequences are there?
• Exactly solvable models of growing interfaces and lattice gases: the Arcetri models, ageing and logarithmic sub-ageing
• Current fluctuations for totally asymmetric exclusion on the relaxation scale
• Kardar-Parisi-Zhang universality class in the synchronization of oscillator lattices with time-dependent noise
• Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [SU (2) X SU(2) X U(1)]/Z_2^2$ symmetry
• KPZ fluctuations in finite volume
• Kardar-Parisi-Zhang fluctuations in the synchronization dynamics of limit-cycle oscillators
• Transition Probabilities of the Bethe Ansatz Solvable Interacting Particle Systems
• Exclusion model of mRNA translation with collision-induced ribosome drop-off
• First-Passage Percolation under extreme disorder: from bond-percolation to Kardar-Parisi-Zhang universality
• Logarithmic superdiffusion in two dimensional driven lattice gases
• Condensation induced by coupled transport processes
• Large-scale kinetic roughening behavior of coffee-ring fronts
• Hydrodynamic approximations for driven dense colloidal mixtures in narrow pores
• Critical Behavior of Non-Intersecting Brownian Motions
• Critical dynamics of the jamming transition in one-dimensional nonequilibrium lattice-gas models
• Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms
• Surface growth on treelike lattices and the upper critical dimension of the KPZ class
• A Unified Interface Model for Dissipative Transport of Bosons and Fermions
• Microscopic fluctuations in the spreading fronts of circular wetting liquid droplets
• Transverse Temperature Interfaces in the Katz-Lebowitz-Spohn Driven Lattice Gas
• Zone clearance in an infinite TASEP with a step initial condition
• From the Riemann surface of TASEP to ASEP
• Crystal Growth Inside an Octant
• Importance Sampling for counting statistics in one-dimensional systems
• Correlation Functions for a Chain of Short Range Oscillators
• Width scaling of an interface constrained by a membrane
• Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time
• Spacings - An Example for Universality in Random Matrix Theory
• Inviscid limit of the active interface equations
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• Universal scaling and criticality of extremes in random matrix theory
• Finite-temperature transport in the gapped spin-1/2 XXZ chain and one-dimensional lattice spinless fermion model
• Influence of coupling symmetries and noise on the critical dynamics of synchronizing oscillator lattices
• Spin correlation functions and quasiparticle decay
• On ASEP with Periodic Step Bernoulli Initial Condition
• Local resetting with geometric confinement
• Current fluctuations in the symmetric zero-range process below and at critical density
• Tuning of friction noise by accessing the rolling-sliding option
• The role of the non-linearity in controlling the surface roughness in the one-dimensional Kardar--Parisi--Zhang growth process
• On the gain of entrainment in the $n$-dimensional ribosome flow model
• Nonconserved critical dynamics of the two-dimensional Ising model as a surface kinetic roughening process
• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
• Dimensional crossover in Kardar-Parisi-Zhang growth
• Weak pinning and long-range anticorrelated motion of phase boundaries in driven diffusive systems
• Integrability in out-of-equilibrium systems
• Universal interface fluctuations in the contact process
• Exponents and front fluctuations in the quenched Kardar-Parisi-Zhang universality class of one- and two- dimensional interfaces
• Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments
• A novel Recurrence-Transience transition and Tracy-Widom growth in a cellular automaton with quenched noise
• The Tracy-Widom distribution at large Dyson index
• Negative feedback and oscillations in a model for mRNA translation
• Random matrix statistics and safety rest areas on interstates in the United States