KPZ fluctuations in finite volume

Abstract

These lecture notes, adapted from the habilitation thesis of the author, survey in a first part various exact results obtained in the past few decades about KPZ fluctuations in one dimension, with a special focus on finite volume effects describing the relaxation to its stationary state of a finite system starting from a given initial condition. The second part is more specifically devoted to an approach allowing to express in a simple way the statistics of the current in the totally asymmetric simple exclusion process in terms of a contour integral on a compact Riemann surface, whose infinite genus limit leads to KPZ fluctuations in finite volume.

Adapted from the habilitation thesis of the author. 99 pages, 25 figures, 605 references

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• An Exact Formula for the Statistics of the Current in the TASEP with Open Boundaries
• Bethe Ansatz Solution for a Defect Particle in the Asymmetric Exclusion Process
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• Spectral gap of the totally asymmetric exclusion process at arbitrary filling
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• The KPZ Limit of ASEP with Boundary
• Transition probabilities and dynamic structure factor in the ASEP conditioned on strong flux
• Exact Non-Stationary Probabilities in the Asymmetric Exclusion Process on a Ring
• Large Deviation Function of the Partially Asymmetric Exclusion Process
• Asymmetric Exclusion Model and Weighted Lattice Paths
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• Microscopic structure of shocks and antishocks in the ASEP conditioned on low current
• Exact solution of an exclusion process with three classes of particles and vacancies
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• Two-Species Asymmetric Simple Exclusion Process with Open Boundaries
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• Coupled Kardar-Parisi-Zhang Equations in One Dimension
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• Current Fluctuations in the exclusion process and Bethe Ansatz
• The Asymmetric Avalanche Process
• The height distribution of the KPZ equation with sharp wedge initial condition: numerical evaluations
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• Tree structures for the current fluctuations in the exclusion process
• The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion
• Universality in dynamical phase transitions of diffusive systems
• Bethe Ansatz and Q-operator for the open ASEP
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• Time-time covariance for last passage percolation with generic initial profile
• An Inhomogeneous Multispecies TASEP on a Ring
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• Reflected Brownian motions in the KPZ universality class
• Bethe ansatz and current distribution for the TASEP with particle-dependent hopping rates
• Bosonization, vicinal surfaces, and hydrodynamic fluctuation theory
• Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface
• On the asymmetric simple exclusion process with multiple species
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• Out-of-equilibrium relaxation of the Edwards-Wilkinson elastic line
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• Kardar-Parisi-Zhang Interfaces with Inward Growth
• Multispecies TASEP and the tetrahedron equation
• Weakly asymmetric bridges and the KPZ equation
• Geometric aspects of the ODE/IM correspondence
• Morphology and scaling in the noisy Burgers equation: Soliton approach to the strong coupling fixed point
• Fluctuations and skewness of the current in the partially asymmetric exclusion process
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Spectral Degeneracies in the Totally Asymmetric Exclusion Process
• Attempted Bethe ansatz solution for one-dimensional directed polymers in random media
• Half-space stationary Kardar-Parisi-Zhang equation
• Crossover to the stochastic Burgers equation for the WASEP with a slow bond
• Unbinding transition in semi-infinite two-dimensional localized systems
• TASEP on a ring in sub-relaxation time scale
• Dynamical Transition in the Open-boundary Totally Asymmetric Exclusion Process
• Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
• Two-point generating function of the free energy for a directed polymer in a random medium
• Determinant solution for the Totally Asymmetric Exclusion Process with parallel update II. Ring geometry
• Kardar-Parisi-Zhang Universality of the Nagel-Schreckenberg Model
• Matrix product solution to an inhomogeneous multi-species TASEP
• Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability
• Short time large deviations of the KPZ equation
• On the equal time two-point distribution of the one-dimensional KPZ equation by replica
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• One-sided reflected Brownian motions and the KPZ fixed point
• Combinatorics of the two-species ASEP and Koornwinder moments
• Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries
• How flat is flat in random interface growth?
• Lectures notes on compact Riemann surfaces
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states
• Asymmetric XXZ chain at the antiferromagnetic transition: Spectra and partition functions
• Large Deviations For Sticky Brownian Motions
• The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions
• KPZ physics and phase transition in a classical single random walker under continuous measurement
• Topology and the Kardar-Parisi-Zhang universality class
• A Riemann-Hilbert approach to the lower tail of the KPZ equation
• Random geometry and the Kardar-Parisi-Zhang universality class
• Emergence of jams in the generalized totally asymmetric simple exclusion process
• Integrable boundary conditions for multi-species ASEP
• Continued Fractions and the Partially Asymmetric Exclusion Process
• Tail of the two-time height distribution for KPZ growth in one dimension
• Power Spectra of the Total Occupancy in the Totally Asymmetric Simple Exclusion Process
• KP governs random growth off a one dimensional substrate
• Superdiffusion from nonabelian symmetries in nearly integrable systems
• Determinant representation for some transition probabilities in the TASEP with second class particles
• Riemann surfaces for KPZ with periodic boundaries
• Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
• Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model
• Symmetries and zero modes in sample path large deviations
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Entanglement entropy growth in stochastic conformal field theory and the KPZ class
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues
• Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces
• Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries
• Kardar-Parisi-Zhang Physics in the Density Fluctuations of Localized Two-Dimensional Wave Packets
• Matrix product solution to a 2-species TASEP with open integrable boundaries
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
• Fluctuations of the winding number of a directed polymer in a random medium
• Supremum of the Airy2 process minus a parabola on a half line
• TASEP hydrodynamics using microscopic characteristics
• Correlations in the Multispecies TASEP and a Conjecture by Lam
• A Determinantal Formula for Catalan Tableaux and TASEP Probabilities
• Hydrodynamic large deviations of TASEP
• Stochastic Fusion of Interacting Particle Systems and Duality Functions
• Universal fluctuation of the average height in the early-time regime of the one-dimensional Kardar-Parisi-Zhang-type growth
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• The exact phase diagram for a class of multispecies asymmetric exclusion processes
• Circular Kardar-Parisi-Zhang interfaces evolving out of the plane
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Nonequilibrium dynamics of a growing interface
• Extrapolation methods and Bethe ansatz for the asymmetric exclusion process
• Combinatorial mappings of exclusion processes
• Critical resonance in the non-intersecting lattice path model
• Relaxation rate of the reverse biased asymmetric exclusion process
• Numerical calculation of three-point branched covers of the projective line
• KPZ statistics of second class particles in ASEP via mixing
• Equivalent T-Q relations and exact results for the open TASEP
• Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica
• Current large deviations for partially asymmetric particle systems on a ring
• $N$-point free energy distribution function in one dimensional random directed polymers
• Long time asymptotics of the totally asymmetric simple exclusion process
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Hidden symmetries in the asymmetric exclusion process
• Multi-state asymmetric simple exclusion processes
• Stationary measures of the KPZ equation on an interval from Enaud-Derrida's matrix product ansatz representation
• Tableaux combinatorics of the two-species PASEP
• Exact Formulas of the Transition Probabilities of the Multi-Species Asymmetric Simple Exclusion Process
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• Matrix product solution to multi-species ASEP with open boundaries
• Generalized matrix Ansatz in the multispecies exclusion process - partially asymmetric case
• Riemann surface for TASEP with periodic boundaries
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Lectures on nonlinear integrable equations and their solutions
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Extreme statistics of non-intersecting Brownian paths
• Hydrodynamic behavior of the 2-TASEP
• Integrability in the weak noise theory
• KPZ universality class and the anchored Toom interface
• Anomalous Hydrodynamics in One Dimensional Electronic Fluid
• Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process
• Spectral signatures of symmetry-breaking dynamical phase transitions
• The Long ans Short Time Asymptotics of the Two-Time Distribution in Local Random Growth
• Fluctuation Exponents of the KPZ equation on a large torus
• Ergodicity and synchronization of the Kardar-Parisi-Zhang equation
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• A direct proof of Kim's identities
• Dynamical Critical Exponent for Two-Species Totally Asymmetric Diffusion on a Ring
• Family of Commuting Operators for the Totally Asymmetric Exclusion Process
• Limiting current distribution for a two species asymmetric exclusion process
• Midpoint distribution of directed polymers in the stationary regime: exact result through linear response
• Differential equations for the KPZ and periodic KPZ fixed points
• Infinite geodesics, competition interfaces and the second class particle in the scaling limit
• Markov duality and Bethe ansatz formula for half-line open ASEP
• High temperature behaviors of the directed polymer on a cylinder
• Enumeration of Corners in Tree-like Tableaux
• Integrability of two-species partially asymmetric exclusion processes
• Current fluctuations for totally asymmetric exclusion on the relaxation scale
• Multiline queues with spectral parameters
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Current fluctuations in a partially asymmetric simple exclusion process with a defect particle
• Probability distributions of multi-species q-TAZRP and ASEP as double cosets of parabolic subgroups
• Short-time large deviations of the spatially averaged height of a KPZ interface on a ring
• The strong Feller property of the open KPZ equation
• Large deviations and fluctuations of real eigenvalues of elliptic random matrices
• A tourist's guide to regularity structures and singular stochastic PDEs
• Riemann surfaces for integer counting processes
• Mixing times for the TASEP on the circle
• Current statistics in the q-boson zero range process
• The combinatorics of hopping particles and positivity in Markov chains
• Epithelial Tissue Growth Dynamics: Universal or Not?
• Integrable fluctuations in the KPZ universality class
• KPZ limit theorems
• Derivation of coupled KPZ equations from interacting diffusions driven by a single-site potential
• Ergodicity results for the open KPZ equation
• From the Riemann surface of TASEP to ASEP
• TASEP Speed Process: An Effective Medium Approach
• Fluctuations of random Motzkin paths II
• Newman-Rivlin asymptotics for partial sums of power series
• From ABC to KPZ
• Representations of Hecke algebras and Markov dualities for interacting particle systems
• Random permutations and related topics
• Random Growth Models
• Contour Integral Formulas for PushASEP on the Ring
• On energy solutions to stochastic Burgers equation
• Polynuclear growth and the Toda lattice
• Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains
• The phase diagram for a class of multispecies permissive asymmetric exclusion processes
• The spectral boundary of the Asymmetric Simple Exclusion Process (ASEP) -- free fermions, Bethe ansatz and random matrix theory

Cited by in corpus (4)

• Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Approach to stationarity for the KPZ fixed point with boundaries
• Current fluctuations for the second class particle : joint statistics