June 2023 A variational formula for large deviations in first-passage percolation under tail estimates
Clément Cosco, Shuta Nakajima
Author Affiliations +
Ann. Appl. Probab. 33(3): 2103-2135 (June 2023). DOI: 10.1214/22-AAP1861


Consider the first passage percolation on the d-dimensional lattice Zd with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations P(T(0,nx)>n(μ+ξ)), for ξ>0 and x0 with a time constant μ, for weights that satisfy a tail assumption P(τe>t)βexp(αtr). When r1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as exp((2dαξr+o(1))n). When 1<rd, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. The case r=d is critical and logarithmic corrections appear. For r(1,d), we show that the large deviation event {T(0,nx)>n(μ+ξ)} is described by a localization of high weights around the endpoints. The picture changes for rd where the configuration is not anymore localized.

Funding Statement

The first author acknowledges that this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 692452).
The second author is partially supported by JSPS KAKENHI 19J00660 and SNSF Grant 176918.


We thank Ofer Zeitouni for suggesting Theorem 1.12. We thank the referee for his valuable comments. We also thank Bobo Hua and Florian Schweiger for helpful discussions. The second author would like to thank Ryoki Fukushima introducing [8], Lemma 3.1.


Download Citation

Clément Cosco. Shuta Nakajima. "A variational formula for large deviations in first-passage percolation under tail estimates." Ann. Appl. Probab. 33 (3) 2103 - 2135, June 2023. https://doi.org/10.1214/22-AAP1861


Received: 1 April 2021; Revised: 1 January 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583666
zbMATH: 07692313
Digital Object Identifier: 10.1214/22-AAP1861

Primary: 60K37
Secondary: 60K35 , 82A51 , 82D30

Keywords: Eden growth model , First-passage percolation , large deviations

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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