Fluctuation properties of the TASEP with periodic initial configuration

Abstract

We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.

33 pages, 4 figure, LaTeX; We added several references to the general framework and techniques used

References in corpus (23)

• Level-Spacing Distributions and the Airy Kernel
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Determinantal Processes and Independence
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Discrete polynuclear growth and determinantal processes
• Exact Solution of the Master Equation for the Asymmetric Exclusion Process
• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Random matrices and determinantal processes
• Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process
• Spatial correlations of the 1D KPZ surface on a flat substrate
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• Matrices coupled in a chain. I. Eigenvalue correlations
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Current Distribution and random matrix ensembles for an integrable asymmetric fragmentation process
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• A determinantal formula for the GOE Tracy-Widom distribution
• Asymmetric Simple Exclusion Process and Modified Random Matrix Ensembles
• Universality for mathematical and physical systems
• Bethe ansatz and current distribution for the TASEP with particle-dependent hopping rates
• One-dimensional stochastic growth and Gaussian ensembles of random matrices
• Fluctuations in the discrete TASEP with periodic initial configurations and the Airy_1 process

Cited by in corpus (144)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• The one-dimensional KPZ equation: an exact solution and its universality
• Growing interfaces uncover universal fluctuations behind scale invariance
• On the Numerical Evaluation of Fredholm Determinants
• The one-dimensional KPZ equation and its universality class
• A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• The KPZ fixed point
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition
• Anisotropic growth of random surfaces in 2+1 dimensions
• Transition between Airy_1 and Airy_2 processes and TASEP fluctuations
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Stationary correlations for the 1D KPZ equation
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• Renormalization fixed point of the KPZ universality class
• Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP
• A stochastic Burgers equation from a class of microscopic interactions
• The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
• Airy processes and variational problems
• Limit process of stationary TASEP near the characteristic line
• Continuum statistics of the Airy2 process
• Local behavior and hitting probabilities of the Airy1 process
• Universality of slow decorrelation in KPZ growth
• Slow decorrelations in KPZ growth
• Finite time corrections in KPZ growth models
• From interacting particle systems to random matrices
• Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
• Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
• The Airy_1 process is not the limit of the largest eigenvalue in GOE matrix diffusion
• Limit processes for TASEP with shocks and rarefaction fans
• On Time Correlations for KPZ Growth in One Dimension
• Two speed TASEP
• Slow relaxation and aging kinetics for the driven lattice gas
• KPZ Scaling Theory and the Semi-discrete Directed Polymer Model
• Airy processes with wanderers and new universality classes
• Determinantal transition kernels for some interacting particles on the line
• A Pfaffian representation for flat ASEP
• Time-time covariance for last passage percolation with generic initial profile
• Point-to-line polymers and orthogonal Whittaker functions
• Scaling limit for Brownian motions with one-sided collisions
• Orbit measures, random matrix theory and interlaced determinantal processes
• Reflected Brownian motions in the KPZ universality class
• Anomalous shock fluctuations in TASEP and last passage percolation models
• Tracy-Widom asymptotics for q-TASEP
• Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Replica approach to the KPZ equation with half Brownian motion initial condition
• The universal Airy_1 and Airy_2 processes in the Totally Asymmetric Simple Exclusion Process
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• TASEP on a ring in sub-relaxation time scale
• Microscopic concavity and fluctuation bounds in a class of deposition processes
• One-sided reflected Brownian motions and the KPZ fixed point
• How flat is flat in random interface growth?
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Revisiting Directed Polymers with heavy-tailed disorder
• Emergence of jams in the generalized totally asymmetric simple exclusion process
• Stationary half-space last passage percolation
• Brownian Motions with One-Sided Collisions: The Stationary Case
• The 1+1 dimensional Kardar-Parisi-Zhang equation: more surprises
• Height fluctuations for the stationary KPZ equation
• KP governs random growth off a one dimensional substrate
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Determinant representation for some transition probabilities in the TASEP with second class particles
• Riemann surfaces for KPZ with periodic boundaries
• Multiplicative functionals on ensembles of non-intersecting paths
• Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model
• Determinantal Correlations for Classical Projection Processes
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Supremum of the Airy2 process minus a parabola on a half line
• TASEP and generalizations: Method for exact solution
• Multi-point distribution of periodic TASEP
• Long time asymptotics of the totally asymmetric simple exclusion process
• Multi-state asymmetric simple exclusion processes
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• Tracy-Widom limit of q-Hahn TASEP
• Matrix product solution to multi-species ASEP with open boundaries
• Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
• Shock fluctuations in flat TASEP under critical scaling
• Distribution of a particle's position in the ASEP with the {alternating} initial condition
• On the partial connection between random matrices and interacting particle systems
• Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison
• Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes
• The transition probability and the probability for the left-most particle's position of the q-TAZRP
• Tilings of non-convex Polygons, skew-Young Tableaux and determinantal Processes
• Formulas for Joint Probabilities for the Asymmetric Simple Exclusion Process
• Interlaced particle systems and tilings of the Aztec diamond
• Asymptotics of Plancherel measures for the infinite-dimensional unitary group
• Finite GUE distribution with cut-off at a shock
• Why random matrices share universal processes with interacting particle systems?
• KPZ universality class and the anchored Toom interface
• Fluctuations in the discrete TASEP with periodic initial configurations and the Airy_1 process
• Universal exit probabilities in the TASEP
• Limiting current distribution for a two species asymmetric exclusion process
• Solution of the Kolmogorov equation for TASEP
• Perturbed GUE Minor Process and Warren's Process with Drifts
• Generalized Green Functions and current correlations in the TASEP
• Green functions for the TASEP with sublattice parallel update
• Determinantal Expressions in Multi-Species TASEP
• Determinantal structures in space inhomogeneous dynamics on interlacing arrays
• Random matrix minor processes related to percolation theory
• KPZ fluctuations in finite volume
• Fluctuations for stationary $q$-TASEP
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• From Vicious Walkers to TASEP
• Moderate deviation and exit time estimates for stationary Last Passage Percolation
• On the exponent governing the correlation decay of the Airy${}_1$ process
• Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point
• TASEP fluctuations with soft-shock initial data
• Transition Probabilities of the Bethe Ansatz Solvable Interacting Particle Systems
• The KPZ fixed point for discrete time TASEPs
• A Unified Interface Model for Dissipative Transport of Bosons and Fermions
• Asymptotics for the ratio and the zeros of multiple Charlier polynomials
• Current statistics in the q-boson zero range process
• Double Interlacing in Random Tiling Models
• Limit distributions for KPZ growth models with spatially homogeneous random initial conditions
• Large time asymptotics of growth models on space-like paths I: PushASEP
• Dimers and orthogonal polynomials: connections with random matrices
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• On the spatial persistence for Airy processes
• Exact solution of interacting particle systems related to random matrices
• GUE $\times$ GUE limit law at hard shocks in ASEP
• On ASEP with Periodic Step Bernoulli Initial Condition
• Point-interacting Brownian motions in the KPZ universality class
• Nonstationary generalized TASEP in KPZ and jamming regimes
• Statistics of overtakes by a tagged agent
• Fluctuations of TASEP on a ring in relaxation time scale
• On the $q$-TASEP with a random initial condition
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Scaling limit of soliton lengths in a multicolor box-ball system
• The oriented swap process and last passage percolation
• Asymptotic fluctuations of geometric q-TASEP, geometric q-PushTASEP and q-PushASEP
• A Finitization of the Bead Process
• TASEP with a general initial condition and a deterministically moving wall
• A convergence framework for Airy${}_{\beta }$ line ensemble via pole evolution
• The half-space Airy stat process
• On the duality between particles and polymers
• Free fermionic probability theory and K-theoretic Schubert calculus
• Multi-point distribution of discrete time periodic TASEP
• Cutoff and discrete Product Structure in ASEP
• Waves in a Spatial Queue: Stop-and-Go at Airport Security
• Non-normal limiting distribution for optimal alignment scores of strings in binary alphabets