Half-space stationary Kardar-Parisi-Zhang equation

Abstract

We study the solution of the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height $h(x,t)$ on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at $x=0$. The boundary condition ${\partial }_{x}h(x,t){|}_{x=0}=A$ corresponds to an attractive wall for $A<0$, and leads to the binding of the polymer to the wall below the critical value $A=-1/2$. Here we choose the initial condition $h(x,0)$ to be a Brownian motion in $x>0$ with drift $-(B+1/2)$. When $A+B\to -1$, the solution is stationary, i.e. $h(\cdot ,t)$ remains at all times a Brownian motion with the same drift, up to a global height shift $h(0,t)$. We show that the distribution of this height shift is invariant under the exchange of parameters $A$ and $B$. For any $A,B>-1/2$, we provide an exact formula characterizing the distribution of $h(0,t)$ at any time $t$, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters $A,B$. In particular, when $(A,B)\to (-1/2,-1/2)$, the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik-Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea-Ferrari-Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.

arXiv comment53 pages

References in corpus (69)

• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• 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
• Growing interfaces uncover universal fluctuations behind scale invariance
• On the Numerical Evaluation of Fredholm Determinants
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• An exact solution for the KPZ equation with flat initial conditions
• Scaling for a one-dimensional directed polymer with boundary conditions
• The one-dimensional KPZ equation and its universality class
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process
• 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
• Exact solution for the stationary Kardar-Parisi-Zhang equation
• The intermediate disorder regime for directed polymers in dimension $1+1$
• Integral Formulas for the Asymmetric Simple Exclusion Process
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• The crossover regime for the weakly asymmetric simple exclusion process
• Stationary correlations for the 1D KPZ equation
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
• Renormalization fixed point of the KPZ universality class
• A determinantal formula for the GOE Tracy-Widom distribution
• Bethe Ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions
• Ground-state properties of one-dimensional ultracold Bose gases in a hard-wall trap
• Directed polymer near a hard wall and KPZ equation in the half-space
• The asymmetric Exclusion Process and Brownian Excursions
• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• Limit process of stationary TASEP near the characteristic line
• Crossover from Growing to Stationary Interfaces in the Kardar-Parisi-Zhang Class
• The KPZ Limit of ASEP with Boundary
• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• Spectral theory for the q-Boson particle system
• Large deviations of the maximal eigenvalue of random matrices
• Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers
• Universal aspects of curved, flat & stationary-state Kardar-Parisi-Zhang statistics
• From interacting particle systems to random matrices
• On ASEP with Step Bernoulli Initial Condition
• Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz
• Pfaffian Schur processes and last passage percolation in a half-quadrant
• Directed random polymers via nested contour integrals
• Memory and universality in interface growth
• Right tail expansion of Tracy-Widom beta laws
• A Pfaffian representation for flat ASEP
• Replica approach to the KPZ equation with half Brownian motion initial condition
• Large fluctuations of the KPZ equation in a half-space
• Attempted Bethe ansatz solution for one-dimensional directed polymers in random media
• From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation
• The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble
• The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Stationary half-space last passage percolation
• Orthogonality of Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• Riemann surfaces for KPZ with periodic boundaries
• When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class
• Replica Bethe ansatz derivation of the GOE Tracy-Widom distribution in one-dimensional directed polymers with free boundary conditions
• Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions
• The asymmetric simple exclusion process with an open boundary
• Crossing probability for directed polymers in random media: exact tail of the distribution

Cited by in corpus (17)

• Measurement-Induced Power-Law Negativity in an Open Monitored Quantum Circuit
• Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment
• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Entanglement Structure and Information Protection in Noisy Hybrid Quantum Circuits
• Noise-induced phase transitions in hybrid quantum circuits
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• 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
• Lyapunov exponents of the half-line SHE
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• KPZ fluctuations in finite volume
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics
• Fishnet four-point integrals: integrable representations and thermodynamic limits
• A stationary model of non-intersecting directed polymers
• Height distributions in interface growth: The role of the averaging process
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach