Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states

Abstract

We consider the spectrum of the totally asymmetric simple exclusion process on a periodic lattice of $L$ sites. The first eigenstates have an eigenvalue with real part scaling as $L^{-3/2}$ for large $L$ with finite density of particles. Bethe ansatz shows that these eigenstates are characterized by four finite sets of positive half-integers, or equivalently by two integer partitions. Each corresponding eigenvalue is found to be equal to the value at its saddle point of a function indexed by the four sets. Our derivation of the large $L$ asymptotics relies on a version of the Euler-Maclaurin formula with square root singularities at both ends of the summation range.

arXiv comment31 pages, 10 figures

References in corpus (62)

• The large deviation approach to statistical mechanics
• Dynamical Ensembles in Nonequilibrium Statistical Mechanics
• Shape Fluctuations and Random Matrices
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Exact Large Deviation Function in the Asymmetric Exclusion Process
• Free-energy distribution of the directed polymer at high temperature
• Growing interfaces uncover universal fluctuations behind scale invariance
• Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Exact Solution of the Master Equation for the Asymmetric Exclusion Process
• A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
• The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
• Dynamic States of a Continuum Traffic Equation with On-Ramp
• Universal cumulants of the current in diffusive systems on a ring
• Exact Spectral Gaps of the Asymmetric Exclusion Process with Open Boundaries
• Integral Formulas for the Asymmetric Simple Exclusion Process
• Exact Current Statistics of the ASEP with Open Boundaries
• Statistical Self-Similarity of One-Dimensional Growth Processes
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• The crossover regime for the weakly asymmetric simple exclusion process
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
• Current large deviation function for the open asymmetric simple exclusion process
• 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
• Spectral gap of the totally asymmetric exclusion process at arbitrary filling
• Exact Non-Stationary Probabilities in the Asymmetric Exclusion Process on a Ring
• Large Deviation Function of the Partially Asymmetric Exclusion Process
• Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions
• A two--parametric family of asymmetric exclusion processes and its exact solution
• Bethe Ansatz calculation of the spectral gap of the asymmetric exclusion process
• Matrix Coordinate Bethe Ansatz: Applications to XXZ and ASEP models
• Some Exact Results for the Exclusion Process
• Exact Solution of the Asymmetric Exclusion Model with Particles of Arbitrary Size
• Total Current Fluctuations in ASEP
• Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
• Tagged Particle Correlations in the Asymmetric Simple Exclusion Process: Finite Size Effects
• Spectrum in multi-species asymmetric simple exclusion process on a ring
• Derivation of a Matrix Product Representation for the Asymmetric Exclusion Process from Algebraic Bethe Ansatz
• Determinant solution for the Totally Asymmetric Exclusion Process with parallel update
• Bethe Ansatz for the Weakly Asymmetric Simple Exclusion Process and phase transition in the current distribution
• Bethe Ansatz and Q-operator for the open ASEP
• Tree structures for the current fluctuations in the exclusion process
• Crossover Scaling Functions in One Dimensional Dynamic Growth Models
• Matrix Ansatz for the Fluctuations of the Current in the ASEP with Open Boundaries
• Algebraic Bethe Ansatz for the two species ASEP with different hopping rates
• Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Karder--Parisi--Zhang-type growth model
• Spectral Degeneracies in the Totally Asymmetric Exclusion Process
• Dynamical Transition in the Open-boundary Totally Asymmetric Exclusion Process
• Determinant solution for the Totally Asymmetric Exclusion Process with parallel update II. Ring geometry
• Singular stochastic PDEs
• The totally asymmetric exclusion process on a ring: Exact relaxation dynamics and associated model of clustering transition
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues
• Universal fluctuation of the average height in the early-time regime of the one-dimensional Kardar-Parisi-Zhang-type growth
• Entropy production and fluctuation relations for a KPZ interface
• Relaxation rate of the reverse biased asymmetric exclusion process
• Hidden symmetries in the asymmetric exclusion process
• Family of Commuting Operators for the Totally Asymmetric Exclusion Process
• Dynamical Critical Exponent for Two-Species Totally Asymmetric Diffusion on a Ring
• Connected Operators for the Totally Asymmetric Exclusion Process

Cited by in corpus (15)

• The KPZ fixed point
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Finite-time fluctuations for the totally asymmetric exclusion process
• Riemann surfaces for KPZ with periodic boundaries
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Extrapolation methods and Bethe ansatz for the asymmetric exclusion process
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process
• Solution of the Kolmogorov equation for TASEP
• KPZ fluctuations in finite volume
• Current fluctuations for totally asymmetric exclusion on the relaxation scale
• Random sparse generators of Markovian evolution and their spectral properties
• Fluctuations of TASEP on a ring in relaxation time scale