English

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

Disordered Systems and Neural Networks 2018-09-24 v2 Statistical Mechanics Mathematical Physics math.MP

Abstract

We consider the fluctuations in the stochastic growth of a one-dimensional interface of height h(x,t)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 t1t_1 and t2>t1t_2>t_1, with droplet initial conditions at t=0t=0. In the limit of large times this JPDF is expected to become a universal function of the time ratio t2/t1t_2/t_1, and of the (properly scaled) heights h(x,t1)h(x,t_1) and h(x,t2)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,t1)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.

Keywords

Cite

@article{arxiv.1804.01948,
  title  = {Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica},
  author = {Jacopo de Nardis and Pierre Le Doussal},
  journal= {arXiv preprint arXiv:1804.01948},
  year   = {2018}
}

Comments

18 pages, typos corrected