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The KPZ Limit of ASEP with Boundary

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Abstract

It was recently proved in Corwin and Shen (CPAM, [CS16]) that under weakly asymmetric scaling, the height functions for ASEP with sources and sinks converges to the Hopf–Cole solution of the KPZ equation with inhomogeneous Neumann boundary conditions. In their assumptions [CS16] chose positive values for the Neumann boundary condition, and they assumed initial data which is close to stationarity. By developing more extensive heat-kernel estimates, we clarify and extend their results to negative values of the Neumann boundary parameters, and we also show how to generalize their results to empty initial data (which is very far from stationarity). Combining our result with Barraquand et al. (Duke Math J, [BBCW17]), we obtain the Laplace transform of the one-point distribution for half-line KPZ, and use this to confirm t1/3-scale GOE Tracy–Widom long-time fluctuations at the origin.

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Acknowledgements

The authorwishes to thank Ivan Corwin for suggesting the problem, for providing helpful discussions about various issues which came up during the writing of the paper, for suggesting the free energy heuristic to obtain the limit and T 1/2-fluctuations in the low-density regime, and also for thoroughly reading the first four preliminary drafts of this paper. We also wish to thank Hao Shen and Li-Cheng Tsai, who provided some very useful discussions. The author was partially supported by the Fernholz Foundation’s “Summer Minerva Fellows” program, as well as summer support from Ivan Corwin’s NSF Grant DMS:1811143.

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  1. Columbia University, New York, USA

    Shalin Parekh

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Communicated by H. Spohn

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Parekh, S. The KPZ Limit of ASEP with Boundary. Commun. Math. Phys. 365, 569–649 (2019). https://doi.org/10.1007/s00220-018-3258-x

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