Convergence of exclusion processes and the KPZ equation to the KPZ fixed point
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- by Jeremy Quastel and Sourav Sarkar
- J. Amer. Math. Soc. 36 (2023), 251-289
- DOI: https://doi.org/10.1090/jams/999
- Published electronically: March 8, 2022
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Abstract:
We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the Kardar-Parisi-Zhang (KPZ) equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of Wu [Tightness and local fluctuation estimates for the KPZ line ensemble, 2021], Dimitrov and Matetski [Ann. Probab. 49 (2021), pp. 2477–2529], the KPZ line ensemble converges to the Airy line ensemble.
For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with $|h(x)|\le C(1+\sqrt {|x|})$ for some $C>0$, or one narrow wedge. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts.
The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates.
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Bibliographic Information
- Jeremy Quastel
- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada
- MR Author ID: 322635
- Email: quastel@math.toronto.edu
- Sourav Sarkar
- Affiliation: University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 1286815
- Email: ss2871@cam.ac.uk
- Received by editor(s): November 23, 2020
- Received by editor(s) in revised form: August 23, 2021, November 5, 2021, and December 1, 2021
- Published electronically: March 8, 2022
- Additional Notes: Both authors were supported by the Natural Sciences and Engineering Research Council of Canada.
- © Copyright 2022 American Mathematical Society
- Journal: J. Amer. Math. Soc. 36 (2023), 251-289
- MSC (2020): Primary 60K35, 60H15; Secondary 82C24
- DOI: https://doi.org/10.1090/jams/999
- MathSciNet review: 4495842