Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica

Abstract

We consider the fluctuations in the stochastic growth of a one-dimensional interface of height $h(x,t)$ described by the Kardar-Parisi-Zhang (KPZ) universality class. We study the joint probability distribution function (JPDF) of the interface heights at two times $t_1$ and $t_2>t_1$, with droplet initial conditions at $t=0$. In the limit of large times this JPDF is expected to become a universal function of the time ratio $t_2/t_1$, and of the (properly scaled) heights $h(x,t_1)$ and $h(x,t_2)$. Using the replica Bethe ansatz method for the KPZ equation, in [J. Stat. Mech. (2017) 053212] we obtained a formula for the JPDF in the (partial) tail regime where $h(x,t_1)$ is large and positive, subsequently found in excellent agreement with experimental and numerical data [Phys. Rev. Lett. 118, 125701 (2017)]. Here we show that our results are in perfect agreement with Johansson's recent rigorous expression of the full JPDF [arXiv:1802.00729 ], thereby confirming the validity of our methods.

arXiv comment18 pages, typos corrected

References in corpus (26)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• 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
• Exact scaling functions for one-dimensional stationary KPZ growth
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• An exact solution for the KPZ equation with flat initial conditions
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• 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 KPZ equation with flat initial condition and the directed polymer with one free end
• Correlation functions of the one-dimensional attractive Bose gas
• The crossover regime for the weakly asymmetric simple exclusion process
• Stationary correlations for the 1D KPZ equation
• Crossover from Growing to Stationary Interfaces in the Kardar-Parisi-Zhang Class
• Memory and universality in interface growth
• Exact formulas for the form factors of local operators in the Lieb-Liniger model
• From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation
• Exact formulas for random growth with half-flat initial data
• Tail of the two-time height distribution for KPZ growth in one dimension
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• On two-time distribution functions in (1+1) random directed polymers
• Characteristic Sign Renewals of Kardar-Parisi-Zhang Fluctuations

Cited by in corpus (4)

• Time-time covariance for last passage percolation with generic initial profile
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• The Long ans Short Time Asymptotics of the Two-Time Distribution in Local Random Growth
• KPZ fluctuations in finite volume