Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica. (English) Zbl 1457.82287
Summary: 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 \(t_2>t\), with droplet initial conditions at \(t = 0\). In the large-time limit, 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. De Nardis and P. Le Doussal, “Tail of the two-time height distribution for KPZ growth in one dimension”, ibid. 2017, No. 5, Article ID 053212, 72 p. (2017; doi:10.1088/1742-5468/aa6bce)] we obtained a formula for the JPDF in the (partial) tail regime where \(h(x,t_1)\) is large and positive, subsequently found to be in excellent agreement with the experimental and numerical data [J. De Nardis et al., “Memory and universality in interface growth”, Phys. Rev. Lett. 118, Article ID 125701, 5 p. (2017; doi:10.1103/PhysRevLett.118.125701)]. Here we show that our results are in perfect agreement with Johansson’s recent rigorous expression of the full JPDF [K. Johansson, “The two-time distribution in geometric last-passage percolation”, Preprint, arXiv:1802.00729], thereby confirming the validity of our methods.
MSC:
| 82C24 | Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics |
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