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

Abstract

Consider a stochastic interface $h(x,t)$, described by the $1+1$ Kardar-Parisi-Zhang (KPZ) equation on the half-line $x\ge 0$. 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.

9 pages, 10 figures

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Cited by in corpus (17)

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• Lower tail of the KPZ equation
• Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions
• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Short time large deviations of the KPZ equation
• Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media
• Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• Lyapunov exponents of the SHE for general initial data
• Lyapunov exponents of the half-line SHE
• Integrability in the weak noise theory
• Short-time large deviations of the spatially averaged height of a KPZ interface on a ring
• Large deviations for the KPZ equation from the KP equation
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth
• High moments of the SHE in the clustering regimes