Steady state of the KPZ equation on an interval and Liouville quantum mechanics

Abstract

We obtain a simple formula for the stationary measure of the height field evolving according to the Kardar-Parisi-Zhang equation on the interval $[0,L]$ with general Neumann type boundary conditions and any interval size. This is achieved using the recent results of Corwin and Knizel (arXiv:2103.12253) together with Liouville quantum mechanics. Our formula allows to easily determine the stationary measure in various limits: KPZ fixed point on an interval, half-line KPZ equation, KPZ fixed point on a half-line, as well as the Edwards-Wilkinson equation on an interval.

Letter + supplementary material. v4: typos corrected, final version

References in corpus (1)

• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics

Cited by in corpus (3)

• Markov limits of steady states of the KPZ equation on an interval
• High temperature behaviors of the directed polymer on a cylinder
• Local KPZ behavior under arbitrary scaling limits