Large deviations for directed percolation on a thin rectangle

Ibrahim, Jean-Paul

ESAIM: Probability and Statistics, Tome 15 (2011), pp. 217-232.

Following the recent investigations of Baik and Suidan in [Int. Math. Res. Not. (2005) 325-337] and Bodineau and Martin in [Electron. Commun. Probab. 10 (2005) 105-112 (electronic)], we prove large deviation properties for a last-passage percolation model in ℤ₊² whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in [J. Baik and T.M. Suidan, Int. Math. Res. Not. (2005) 325-337] and [T. Bodineau and J. Martin, Electron. Commun. Probab. 10 (2005) 105-112 (electronic)], on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.

[1] J. Baik and T.M. Suidan, A GUE central limit theorem and universality of directed first and last passage site percolation. Int. Math. Res. Not. (2005) 325-337. | MR | Zbl

[2] Yu. Baryshnikov, GUEs and queues. Probab. Theory Relat. Fields 119 (2001) 256-274. | MR | Zbl

[3] G. Ben Arous, A. Dembo and A. Guionnet, Aging of spherical spin glasses. Probab. Theory Relat. Fields 120 (2001) 1-67. | MR | Zbl

[4] G. Ben Arous and A. Guionnet, Large deviations for Wigner's law and Voiculescu's non-commutative entropy. Probab. Theory Relat. Fields 108 (1997) 517-542. | MR | Zbl

[5] T. Bodineau and J. Martin, A universality property for last-passage percolation paths close to the axis. Electron. Commun. Probab. 10 (2005) 105-112 (electronic). | EuDML | MR | Zbl

[6] L. Breiman, Probability, Classics in Applied Mathematics 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992). Corrected reprint of the 1968 original. | MR | Zbl

[7] D.L. Burkholder, Distribution function inequalities for martingales. Ann. Probability 1 (1973) 19-42. | MR | Zbl

[8] S. Chatterjee, A simple invariance theorem. Preprint arXiv:math.PR/0508213 (2005).

[9] S. Csörgő and P. Hall, The Komlós-Major-Tusnády approximations and their applications. Austral. J. Statist. 26 (1984) 189-218. | MR | Zbl

[10] B. Davis, On the Lp norms of stochastic integrals and other martingales. Duke Math. J. 43 (1976) 697-704. | MR | Zbl

[11] D. Féral, On large deviations for the spectral measure of discrete coulomb gas, in Séminaire de Probabilités, XLI. Lecture Notes in Math. 1934. Springer, Berlin (2008) 19-50. | MR | Zbl

[12] D.H. Fuk, Certain probabilistic inequalities for martingales. Sibirsk. Mat. Ž. 14 (1973) 185-193, 239. | MR | Zbl

[13] D.H. Fuk and S.V. Nagaev, Probabilistic inequalities for sums of independent random variables. Teor. Verojatnost. i Primenen. 16 (1971) 660-675. | MR | Zbl

[14] J. Gravner, C.A. Tracy and H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 (2001) 1085-1132. | MR | Zbl

[15] K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 (1998) 151-204. | MR | Zbl

[16] K. Johansson, Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. | MR | Zbl

[17] J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrscheinlichkeitstheor. und Verw. Geb. 34 (1976) 33-58. | MR | Zbl

[18] W. König, Orthogonal polynomial ensembles in probability theory. Prob. Surveys 2 (2005) 385-447 (electronic). | MR | Zbl

[19] M. Ledoux, Deviation inequalities on largest eigenvalues, in Geometric aspects of functional analysis. Lecture Notes in Math. 1910 (2007) 167-219. | MR | Zbl

[20] M. Ledoux and B. Rider, Small deviations for beta ensembles. Preprint (2010). | MR | Zbl

[21] M.L. Mehta, Random matrices, 2nd edition. Academic Press Inc., Boston, MA (1991). | MR | Zbl

[22] T. Mikosch and A.V. Nagaev, Large deviations of heavy-tailed sums with applications in insurance. Extremes 1 (1998) 81-110. | MR | Zbl

[23] N. O'Connell and M. Yor, A representation for non-colliding random walks. Electron. Commun. Probab. 7 (2002) 1-12 (electronic). | MR | Zbl

[24] D.l Revuz and M. Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293, 3rd edition,. Springer-Verlag, Berlin (1999). | MR | Zbl

[25] E.B. Saff and V. Totik, Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 316. Springer-Verlag, Berlin (1997). Appendix B by Thomas Bloom. | MR | Zbl

[26] A.I. Sakhanenko, A new way to obtain estimates in the invariance principle, in High dimensional probability, II (Seattle, WA, 1999), Progr. Probab. 47. Birkhäuser Boston, Boston, MA (2000) 223-245. | MR | Zbl

[27] S. Sawyer, A remark on the Skorohod representation. Z. Wahrscheinlichkeitstheor. und Verw. Geb. 23 (1972) 67-74. | MR | Zbl

[28] A.V. Skorokhod, Studies in the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass (1965). | MR | Zbl

[29] T. Suidan, A remark on a theorem of Chatterjee and last passage percolation. J. Phys. A 39 (2006) 8977-8981. | MR | Zbl

[30] C.A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel. Phys. Lett. B 305 (1993) 115-118. | MR | Zbl