Fluctuation exponents of the half-space KPZ at stationarity

TL;DR

This paper investigates the fluctuation behavior of the half-space KPZ equation with stationary initial data, deriving variance identities and obtaining optimal fluctuation exponents across different regimes, along with growth rate analysis.

Contribution

It introduces a variance identity linking KPZ height fluctuations to polymer endpoint fluctuations, providing optimal exponents and boundary parameter dependence.

Findings

Derived variance identity for half-space KPZ with stationary data

Obtained optimal fluctuation exponents in subcritical and critical regimes

Computed average growth rate as a function of boundary parameter

Abstract

We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. Utilizing this identity, we obtain estimates for the polymer endpoints, leading to optimal fluctuation exponents for the height function in both the subcritical and critical regimes, as well as an optimal upper bound for the fluctuation exponents in the extended critical regime. We also compute the average growth rate as a function of the boundary parameter.