Universal fluctuation of the average height in the early-time regime of the one-dimensional Kardar-Parisi-Zhang-type growth

TL;DR

This paper investigates the universal fluctuation behavior of the average height in early-time regimes of 1D KPZ-type growth, deriving scaling forms and confirming universality through analytical and simulation results.

Contribution

It derives the scaling form of height fluctuations in early-time KPZ growth and demonstrates the universality of the scaling functions and large deviation functions.

Findings

Universal scaling functions for cumulants established

Analytic predictions confirmed by simulations

Early-time fluctuation behavior characterized

Abstract

The statistics of the average height fluctuation of the one-dimensional Kardar-Parisi-Zhang(KPZ)-type surface is investigated. Guided by the idea of local stationarity, we derive the scaling form of the characteristic function in the early-time regime, $t\ll N^{3/2}$ with $t$ time and $N$ the system size, from the known characteristic function in the stationary state ($t\gg N^{3/2}$) of the single-step model derivable from a Bethe Ansatz solution, and thereby find the scaling properties of the cumulants and the large deviation function in the early-time regime. These results, combined with the scaling analysis of the KPZ equation, imply the existence of the universal scaling functions for the cumulants and an universal large deviation function. The analytic predictions are supported by the simulation results for two different models.