Electronic Communications in Probability

Upper tail large deviations in Brownian directed percolation

Christopher Janjigian

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This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof of a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 45, 10 pp.

Received: 4 November 2018
Accepted: 19 June 2019
First available in Project Euclid: 5 July 2019

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Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Brownian directed percolation large deviations largest eigenvalues GUE

Creative Commons Attribution 4.0 International License.


Janjigian, Christopher. Upper tail large deviations in Brownian directed percolation. Electron. Commun. Probab. 24 (2019), paper no. 45, 10 pp. doi:10.1214/19-ECP249. https://projecteuclid.org/euclid.ecp/1562292106

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  • [1] Jinho Baik and Toufic M. Suidan, A GUE central limit theorem and universality of directed first and last passage site percolation, Int. Math. Res. Not. (2005), no. 6, 325–337.
  • [2] Yu. Baryshnikov, GUEs and queues, Probab. Theory Related Fields 119 (2001), no. 2, 256–274.
  • [3] G. Ben Arous, A. Dembo, and A. Guionnet, Aging of spherical spin glasses, Probab. Theory Related Fields 120 (2001), no. 1, 1–67.
  • [4] G. Ben Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Related Fields 108 (1997), no. 4, 517–542.
  • [5] Thierry Bodineau and James Martin, A universality property for last-passage percolation paths close to the axis, Electron. Comm. Probab. 10 (2005), 105–112 (electronic).
  • [6] Federico Ciech and Nicos Georgiou, A large deviation principle for last passage times in an asymmetric bernoulli potential, arXiv:1810.11377 (2018).
  • [7] Elnur Emrah, The shape functions of certain exactly solvable inhomogeneous planar corner growth models, Electron. Comm. Probab. 21 (2016), no. 42, 1–16.
  • [8] Elnur Emrah and Chris Janjigian, Large deviations for some corner growth models with inhomogeneity, Markov Process. Related Fields 23 (2017), 267–312.
  • [9] Nicos Georgiou and Timo Seppäläinen, Large deviation rate functions for the partition function in a log-gamma distributed random potential, Ann. Probab. 41 (2013), no. 6, 4248–4286.
  • [10] Peter W. Glynn and Ward Whitt, Departures from many queues in series, Ann. Appl. Probab. 1 (1991), no. 4, 546–572.
  • [11] Janko Gravner, Craig A. Tracy, and Harold Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Statist. Phys. 102 (2001), no. 5-6, 1085–1132.
  • [12] B. M. Hambly, James B. Martin, and Neil O’Connell, Concentration results for a Brownian directed percolation problem, Stochastic Process. Appl. 102 (2002), no. 2, 207–220.
  • [13] J. M. Harrison and R. J. Williams, Brownian models of feedforward queueing networks: quasireversibility and product form solutions, Ann. Appl. Probab. 2 (1992), no. 2, 263–293.
  • [14] Jean-Paul Ibrahim, Large deviations for directed percolation on a thin rectangle, ESAIM Probab. Stat. 15 (2011), 217–232.
  • [15] Chris Janjigian, Large deviations of the free energy in the O’Connell-Yor polymer, J. Stat. Phys. 160 (2015), no. 4, 1054–1080.
  • [16] Marek Kuczma, An introduction to the theory of functional equations and inequalities, second ed., Birkhäuser Verlag, Basel, 2009, Edited and with a preface by Attila Gilányi.
  • [17] M. Ledoux, Deviation inequalities on largest eigenvalues, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1910, Springer, Berlin, 2007, pp. 167–219.
  • [18] Neil O’Connell and Marc Yor, Brownian analogues of Burke’s theorem, Stochastic Process. Appl. 96 (2001), no. 2, 285–304.
  • [19] Neil O’Connell and Marc Yor, A representation for non-colliding random walks, Electron. Comm. Probab. 7 (2002), 1–12 (electronic).
  • [20] Firas Rassoul-Agha and Timo Seppäläinen, A course on large deviations with an introduction to Gibbs measures, Graduate Studies in Mathematics, vol. 162, American Mathematical Society, Providence, RI, 2015.
  • [21] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.
  • [22] T. Seppäläinen, Coupling the totally asymmetric simple exclusion process with a moving interface, Markov Process. Related Fields 4 (1998), no. 4, 593–628, I Brazilian School in Probability (Rio de Janeiro, 1997).
  • [23] Timo Seppäläinen, Large deviations for increasing sequences on the plane, Probab. Theory Related Fields 112 (1998), no. 2, 221–244.
  • [24] Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174.