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Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point

Tomer Asida^*, Eli Livne^†, and Baruch Meerson^‡

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

^*tomer.asida@mail.huji.ac.il
^†livne@phys.huji.ac.il
^‡meerson@mail.huji.ac.il

Phys. Rev. E 99, 042132 – Published 22 April, 2019

DOI: https://doi.org/10.1103/PhysRevE.99.042132

Abstract

We consider a stochastic interface $h (x, t)$ , described by the $1 + 1$ Kardar-Parisi-Zhang (KPZ) equation on the half line $x \geq 0$ with the reflecting boundary at $x = 0$ . The interface is initially flat, $h (x, t = 0) = 0$ . We focus on the short-time probability distribution $P (H, L, t)$ of the height $H$ of the interface at point $x = L$ . Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as $- \sqrt{t} ln P ≃ {| H |}^{3 / 2} f_{-} (L / \sqrt{| H | t})$ and calculate the function $f_{-}$ analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of $L$ , $L_{c} = 0.602 23 \dots \sqrt{| H | t}$ . The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as $- \sqrt{t} ln P ≃ {| H |}^{5 / 2} f_{+} (L / \sqrt{| H | t})$ . We evaluate the function $f_{+}$ using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value $L_{c} ≃ 2 \sqrt{2 | H | t} / π$ . This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, $5 / 2$ . It is smoothed, however, by small diffusion effects.

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Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point

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