Short-time large deviations of the spatially averaged height of a KPZ interface on a ring

Abstract

Using the optimal fluctuation method, we evaluate the short-time probability distribution $P(\bar{H},L,t=T)$ of the spatially averaged height $\bar{H}=(1/L){\int }_0^{L}h(x,t=T)dx$ of a one-dimensional interface $h(x,t)$ governed by the Kardar-Parisi-Zhang equation $${\partial }_{t}h=\nu {\partial }_x^{2}h+\frac{\lambda }{2}{\left({\partial }_{x}h\right)}^2+\sqrt{D}\xi (x,t)$$ on a ring of length $L$. The process starts from a flat interface, $h(x,t=0)=0$. Both at $\lambda \bar{H}<0$, and at sufficiently small positive $\lambda \bar{H}$ the optimal (that is, the least-action) path $h(x,t)$ of the interface, conditioned on $\bar{H}$, is uniform in space, and the distribution $P(\bar{H},L,T)$ is Gaussian. However, at sufficiently large $\lambda \bar{H}>0$ the spatially uniform solution becomes sub-optimal and gives way to non-uniform optimal paths. We study them, and the resulting non-Gaussian distribution $P(\bar{H},L,T)$, analytically and numerically. The loss of optimality of the uniform solution occurs via a dynamical phase transition of either first, or second order, depending on the rescaled system size $\ell =L/\sqrt{\nu T}$, at a critical value $\bar{H}={\bar{H}}_{\text{c}}(\ell )$. At large but finite $\ell$ the transition is of first order. Remarkably, it becomes an "accidental" second-order transition in the limit of $\ell \to \infty$, where a large-deviation behavior $-\ln P(\bar{H},L,T)\simeq (L/T)f(\bar{H})$ (in the units $\lambda =\nu =D=1$) is observed. At small $\ell$ the transition is of second order, while at $\ell =O(1)$ transitions of both types occur.

22 pages, 13 figures

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