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

TL;DR

This paper analyzes the short-time probability distribution of the spatially averaged height of a KPZ interface on a ring, revealing a dynamical phase transition between uniform and non-uniform optimal paths depending on system size and height.

Contribution

It provides an analytical and numerical study of the phase transition in the optimal path structure of the KPZ interface's height distribution, including the first and second order transitions.

Findings

Identifies a dynamical phase transition in the optimal path structure.

Shows the transition changes order depending on system size.

Derives the large-deviation behavior of the height distribution.

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 $$ \partial_th=\nu \partial_x^2h+\frac{\lambda}{2} \left(\partial_xh\right)^2+\sqrt{D}\xi(x,t) $$ 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.