Large deviations for directed percolation on a thin rectangle

Jean-Paul Ibrahim

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 217-232
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Ibrahim, Jean-Paul. "Large deviations for directed percolation on a thin rectangle." ESAIM: Probability and Statistics 15 (2012): 217-232. <http://eudml.org/doc/222485>.

@article{Ibrahim2012,
abstract = {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.},
author = {Ibrahim, Jean-Paul},
journal = {ESAIM: Probability and Statistics},
keywords = {Large deviations; random growth model; Skorokhod embedding theorem; large deviations},
language = {eng},
month = {1},
pages = {217-232},
publisher = {EDP Sciences},
title = {Large deviations for directed percolation on a thin rectangle},
url = {http://eudml.org/doc/222485},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Ibrahim, Jean-Paul
TI - Large deviations for directed percolation on a thin rectangle
JO - ESAIM: Probability and Statistics
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 217
EP - 232
AB - 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.
LA - eng
KW - Large deviations; random growth model; Skorokhod embedding theorem; large deviations
UR - http://eudml.org/doc/222485
ER -

References

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  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.  
  2. Yu. Baryshnikov, GUEs and queues. Probab. Theory Relat. Fields119 (2001) 256–274.  
  3. G. Ben Arous, A. Dembo and A. Guionnet, Aging of spherical spin glasses. Probab. Theory Relat. Fields120 (2001) 1–67.  
  4. G. Ben Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields108 (1997) 517–542.  
  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).  
  6. L. Breiman, Probability, Classics in Applied Mathematics7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992). Corrected reprint of the 1968 original.  
  7. D.L. Burkholder, Distribution function inequalities for martingales. Ann. Probability1 (1973) 19–42.  
  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.  
  10. B. Davis, On the Lp norms of stochastic integrals and other martingales. Duke Math. J.43 (1976) 697–704.  
  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.  
  12. D.H. Fuk, Certain probabilistic inequalities for martingales. Sibirsk. Mat. Ž.14 (1973) 185–193, 239.  
  13. D.H. Fuk and S.V. Nagaev, Probabilistic inequalities for sums of independent random variables. Teor. Verojatnost. i Primenen.16 (1971) 660–675.  
  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.  
  15. K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J.91 (1998) 151–204.  
  16. K. Johansson, Shape fluctuations and random matrices. Comm. Math. Phys.209 (2000) 437–476.  
  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.  
  18. W. König, Orthogonal polynomial ensembles in probability theory. Prob. Surveys2 (2005) 385–447 (electronic).  
  19. M. Ledoux, Deviation inequalities on largest eigenvalues, in Geometric aspects of functional analysis. Lecture Notes in Math.1910 (2007) 167–219.  
  20. M. Ledoux and B. Rider, Small deviations for beta ensembles. Preprint (2010).  
  21. M.L. Mehta, Random matrices, 2nd edition. Academic Press Inc., Boston, MA (1991).  
  22. T. Mikosch and A.V. Nagaev, Large deviations of heavy-tailed sums with applications in insurance. Extremes1 (1998) 81–110.  
  23. N. O’Connell and M. Yor, A representation for non-colliding random walks. Electron. Commun. Probab.7 (2002) 1–12 (electronic).  
  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).  
  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.  
  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.  
  27. S. Sawyer, A remark on the Skorohod representation. Z. Wahrscheinlichkeitstheor. und Verw. Geb.23 (1972) 67–74.  
  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).  
  29. T. Suidan, A remark on a theorem of Chatterjee and last passage percolation. J. Phys. A39 (2006) 8977–8981.  
  30. C.A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel. Phys. Lett. B305 (1993) 115–118.  

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