Fluctuation exponents of the open KPZ equation in the maximal current phase

TL;DR

This paper investigates the fluctuation behavior of the height function in the open KPZ equation within the maximal current phase, establishing bounds on variance growth for specific spatial scaling regimes.

Contribution

It provides the first bounds on the variance of the height function for the open KPZ equation in the maximal current phase, combining techniques from periodic KPZ analysis and Gibbsian line ensembles.

Findings

Established bounds on variance for height function in maximal current phase.

Identified fluctuation exponents for the open KPZ equation with stationary initial conditions.

Extended methods from periodic KPZ to open boundary conditions.

Abstract

We consider the open KPZ equation $H(x,t)$ on the interval $[0,L]$ with Neumann boundary conditions depending on parameters $u,v\ge 0$ (the so-called maximal current phase). For $L \sim t^{\alpha}$ and stationary initial conditions, we obtain matching upper and lower bounds on the variance of the height function $H(0,t)$ for $\alpha \in [0,\frac23]$. Our proof combines techniques from arXiv:2111.03650, which treated the periodic KPZ equation, with Gibbsian line ensemble methods based on the probabilistic structure of the stationary measures developed in arXiv:2103.12253, arXiv:2105.15178, arXiv:2105.03946, arXiv:2306.05983, arXiv:2404.13444.