Large deviations of the interface height in the Golubovi\'{c}-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 Golubovi\'{c}-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.