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Local Behaviour of Airy Processes

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Abstract

The Airy processes describe limit fluctuations in a wide range of growth models, where each particular Airy process depends on the geometry of the initial profile. We show how the coupling method, developed in the last-passage percolation context, can be used to prove existence of a continuous version and local convergence to Brownian motion. By using similar arguments, we further extend these results to a two parameter limit fluctuation process (Airy sheet).

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Notes

  1. The coupling method is also an useful tool to study the long time behaviour of the KPZ fixed point and prove ergodicity [17, 21] of its local increments.

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Acknowledgements

The author would like to thank Eric Cator, Ivan Corwin and Jeremy Quastel for enlightening discussions, and Patrik Ferrari and Timo Seppäläinen for useful comments and corrections of a previous version of this article. The author was partially supported by the CNPQ Grants 421383/2016-0 and 302830/2016-2, and by the FAPERJ Grant E-26/203.048/2016.

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  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, Caixa Postal 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

    Leandro P. R. Pimentel

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Correspondence to Leandro P. R. Pimentel.

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P. R. Pimentel, L. Local Behaviour of Airy Processes. J Stat Phys 173, 1614–1638 (2018). https://doi.org/10.1007/s10955-018-2147-1

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