Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth

Abstract

We study large deviations of the one-point height distribution, $\mathcal{P}(H,T)$, of a stochastic interface, governed by the Golubović-Bruinsma equation $${\partial }_{t}h=-\nu {\partial }_x^{4}h+\frac{\lambda }{2}{\left({\partial }_{x}h\right)}^2+\sqrt{D}\xi (x,t),$$ where $h(x,t)$ is the interface height at point $x$ and time $t$, and $\xi (x,t)$ is the Gaussian white noise. The interface is initially flat, and $H$ is defined by the relation $h(x=0,t=T)=H$. Using the optimal fluctuation method (OFM), we focus on the short-time limit. Here the typical fluctuations of $H$ are Gaussian, and we evaluate the strongly asymmetric and non-Gaussian tails of $\mathcal{P}(H,T)$. We show that the upper tail scales as $-\ln \mathcal{P}(H,T)\sim H^{11/6}/T^{5/6}$. The lower tail, which scales as $-\ln \mathcal{P}(H,T)\sim H^{5/2}/T^{1/2}$, coincides with its counterpart for the Kardar-Parisi-Zhang equation, and we uncover a simple physical mechanism behind this universality. Finally, we verify our asymptotic results for the tails, and compute the large deviation function of $H$, numerically.

8 pages, 8 figures

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