Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line

TL;DR

This paper analyzes the short-time probability distribution of the height of a KPZ interface on a half-line with boundary conditions, revealing different scaling regimes and asymptotic behaviors.

Contribution

It introduces a comprehensive scaling analysis of the KPZ interface on a half-line, uncovering new asymptotic regimes and boundary effects on the height distribution.

Findings

Scaling of the probability distribution depends on the boundary parameter A.

At small and moderate A, the distribution reduces to a known simple scaling.

For large A, two asymptotic regimes are identified with distinct scaling behaviors.

Abstract

Consider a stochastic interface $h(x,t)$, described by the $1+1$ Kardar-Parisi-Zhang (KPZ) equation on the half-line $x\geq 0$. The interface is initially flat, $h(x,t=0)=0$, and driven by a Neumann boundary condition $\partial_x h(x=0,t)=A$ and by the noise. We study the short-time probability distribution $\mathcal{P}\left(H,A,t\right)$ of the one-point height $H=h(x=0,t)$. Using the optimal fluctuation method, we show that $-\ln \mathcal{P}\left(H,A,t\right)$ scales as $t^{-1/2} s \left(H,A t^{1/2}\right)$. For small and moderate $|A|$ this more general scaling reduces to the familiar simple scaling $-\ln \mathcal{P}\left(H,A,t\right)\simeq t^{-1/2} s(H)$, where $s$ is independent of $A$ and time and equal to one half of the corresponding large-deviation function for the full-line problem. For large $|A|$ we uncover two asymptotic regimes. At very short time the simple scaling is restored, whereas at intermediate times the scaling remains more general and $A$-dependent. The distribution tails, however, always exhibit the simple scaling in the leading order.