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.

arXiv commentLetter + supplementary material. v4: typos corrected, final version

References in corpus (20)

• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• Solving the Schwarzian via the Conformal Bootstrap
• Exponential functionals of Brownian motion, I: Probability laws at fixed time
• Renormalization fixed point of the KPZ universality class
• The asymmetric Exclusion Process and Brownian Excursions
• DIFFUSION IN ONE DIMENSIONAL RANDOM MEDIUM AND HYPERBOLIC BROWNIAN MOTION
• An Effective Theory for Midgap States in Doped Spin Ladder and Spin-Peierls Systems: Liouville Quantum Mechanics
• Pfaffian Schur processes and last passage percolation in a half-quadrant
• Directed random polymers via nested contour integrals
• Localization of thermal packets and metastable states in Sinai model
• Strong clustering of non-interacting, passive sliders driven by a Kardar-Parisi-Zhang surface
• Half-space stationary Kardar-Parisi-Zhang equation
• Liouville Quantum Mechanics on a Lattice Large from Geometry of Quantum Lorentz Group
• Thermal Equilibrium with the Wiener Potential: Testing the Replica Variational Approximation
• Asymmetric Simple Exclusion Process with open boundaries and Quadratic Harnesses
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries
• The spatial statistical properties of wave functions in a disordered finite one-dimensional sample
• Dual representations of Laplace transforms of Brownian excursion and generalized meanders
• Scaling of critical wavefunctions at topological Anderson transitions in 1D

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
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