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2022 Second order cubic corrections of large deviations for perturbed random walks
Giancarlos Oviedo, Gonzalo Panizo, Alejandro F. Ramírez
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Electron. J. Probab. 27: 1-45 (2022). DOI: 10.1214/22-EJP786

Abstract

We prove that the Beta random walk, introduced in [BC17] 2017, has cubic fluctuations from the large deviation principle of the GUE Tracy-Widom type for arbitrary values α>0 and β>0 of the parameters of the Beta distribution, removing previous restrictions on their values. Furthermore, we prove that the GUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. We also show that any random walk in space-time random environment that matches certain moments with the Beta random walk also has GUE Tracy-Widom fluctuations in the intermediate disorder regime. As a corollary we show the emergence of GUE Tracy-Widom fluctuations from the large deviation principle for trajectories ending at boundary points for random walks in space (time-independent) i.i.d. Dirichlet random environment in dimension d=2 for a class of asymptotic behavior of the parameters.

Funding Statement

Alejandro F. Ramírez has been partially supported by Iniciativa Científica Milenio and by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1180259 and 1220396.

Acknowledgments

Alejandro F. Ramírez is grateful to José A. Ramírez for useful discussions.

Citation

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Giancarlos Oviedo. Gonzalo Panizo. Alejandro F. Ramírez. "Second order cubic corrections of large deviations for perturbed random walks." Electron. J. Probab. 27 1 - 45, 2022. https://doi.org/10.1214/22-EJP786

Information

Received: 16 August 2021; Accepted: 27 April 2022; Published: 2022
First available in Project Euclid: 10 May 2022

Digital Object Identifier: 10.1214/22-EJP786

Subjects:
Primary: 60K37 , 82C23 , 82C41 , 82D30

Keywords: beta random walk , GUE Tracy-Widom distribution , Random walk in random environment

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