Replica approach to the KPZ equation with half Brownian motion initial condition

Abstract

We consider the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We employ the replica Bethe ansatz and show that the generating function of the exponential moments of the height is expressed as a Fredholm determinant. From this the height distribution and its asymptotics are studied. Furthermore using the replica method we also discuss the multi-point height distribution. We find that some nice properties of the deformed Airy functions play an important role in the analysis.

37 pages, 2 figures

References in corpus (11)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• An exact solution for the KPZ equation with flat initial conditions
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• The One-dimensional KPZ Equation and the Airy Process
• From interacting particle systems to random matrices
• The height distribution of the KPZ equation with sharp wedge initial condition: numerical evaluations
• Two-point generating function of the free energy for a directed polymer in a random medium
• Polynuclear growth model, GOE${}^2$ and random matrix with deterministic source

Cited by in corpus (24)

• Exact solution for the stationary Kardar-Parisi-Zhang equation
• Stationary correlations for the 1D KPZ equation
• Directed polymer near a hard wall and KPZ equation in the half-space
• Spectral theory for the q-Boson particle system
• Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions
• Directed random polymers via nested contour integrals
• A Pfaffian representation for flat ASEP
• Short time growth of a KPZ interface with flat initial conditions
• Exact formulas for the form factors of local operators in the Lieb-Liniger model
• Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation
• Half-space stationary Kardar-Parisi-Zhang equation
• On the equal time two-point distribution of the one-dimensional KPZ equation by replica
• On integrable directed polymer models on the square lattice
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• The propagator of the attractive delta-Bose gas in one dimension
• The crossing probability for directed polymers in random media
• Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Crossing probability for directed polymers in random media: exact tail of the distribution
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz