The KPZ Limit of ASEP with Boundary

Abstract

It was recently proved in [Corwin-Shen, 2016] that under weak asymmetry scaling, the height functions for open ASEP on the half-line and on a bounded interval converge to the Hopf-Cole solution of the KPZ equation with Neumann boundary conditions. In their assumptions [Corwin-Shen, 2016] chose positive values for the Neumann boundary conditions, and they assumed initial data which is close to stationarity. By developing more extensive heat-kernel estimates, we extend their results to negative values of the Neumann boundary parameters, and we also show how to generalize their results to narrow-wedge initial data (which is very far from stationarity). As a corollary via [Barraquand-Borodin-Corwin-Wheeler, 2017], we obtain the Laplace transform of the one-point distribution for half-line KPZ, and use this to prove $t^{1/3}$-scale GOE Tracy-Widom long-time fluctuations.

70 pages, 1 figure

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• The KPZ fixed point
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
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Cited by in corpus (27)

• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Lower tail of the KPZ equation
• Markov processes related to the stationary measure for the open KPZ equation
• Half-space Macdonald processes
• Half-space stationary Kardar-Parisi-Zhang equation
• Markov limits of steady states of the KPZ equation on an interval
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• KPZ equation limit of stochastic higher spin six vertex model
• From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
• The lower tail of the half-space KPZ equation
• Lyapunov exponents of the half-line SHE
• KPZ equation limit of sticky Brownian motion
• Some recent progress in singular stochastic PDEs
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• Approximating the stationary distribution of the ASEP with open boundaries
• Markov duality and Bethe ansatz formula for half-line open ASEP
• KPZ fluctuations in finite volume
• High temperature behaviors of the directed polymer on a cylinder
• The KPZ equation and moments of random matrices
• Boundary renormalisation of SPDEs
• Intermediate Disorder regime for half-space directed polymers
• Large deviation bounds for the Airy point process
• Multiplicative SHE limit of random walks in space-time random environments
• Non-Stationary KPZ equation from ASEP with slow bonds
• Scaling limit of a directed polymer among a Poisson field of independent walks
• Stationary measure for six-vertex model on a strip