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Steady state of the KPZ equation on an interval and Liouville quantum mechanics
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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.
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Large time cumulants of the KPZ equation on an interval
Exact large-time cumulants and upper-tail large deviation rate for the open KPZ equation on an interval are derived from replica Bethe ansatz and from a scaling limit of open ASEP.
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