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

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.