## Electronic Communications in Probability

### A Universality Property for Last-Passage Percolation Paths Close to the Axis

#### Abstract

We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d. weights whose common distribution has a finite $(2+p)$th moment. We study the fluctuations of the passage time from the origin to the point $(n,n^a)$. We show that, for suitable $a$ (depending on $p$), this quantity, appropriately scaled, converges in distribution as $n\to\infty$ to the Tracy-Widom distribution, irrespective of the underlying weight distribution. The argument uses a coupling to a Brownian directed percolation problem and the strong approximation of Komlós, Major and Tusnády.

#### Article information

Source
Electron. Commun. Probab., Volume 10 (2005), paper no. 11, 105-112.

Dates
Accepted: 9 June 2005
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058076

Digital Object Identifier
doi:10.1214/ECP.v10-1139

Mathematical Reviews number (MathSciNet)
MR2150699

Zentralblatt MATH identifier
1111.60068

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Bodineau, Thierry; Martin, James. A Universality Property for Last-Passage Percolation Paths Close to the Axis. Electron. Commun. Probab. 10 (2005), paper no. 11, 105--112. doi:10.1214/ECP.v10-1139. https://projecteuclid.org/euclid.ecp/1465058076

#### References

• Baccelli, F.; Borovkov, A.; Mairesse, J. Asymptotic results on infinite tandem queueing networks. Probab. Theory Related Fields 118 (2000), no. 3, 365–405.
• Baik, J.; Ben Arous, G.; Péché, S. Phase transition of the largest eigenvalue for non-null sample covariance matrices. Preprint math.PR/0403022
• Baik, Jinho; Deift, Percy; McLaughlin, Ken T.-R.; Miller, Peter; Zhou, Xin. Optimal tail estimates for directed last passage site percolation with geometric random variables. Adv. Theor. Math. Phys. 5 (2001), no. 6, 1207–1250.
• Baik, J.; Suidan, T. A GUE central limit theorem and universality of directed first and last passage site percolation. Int. Math. Res. Not. 2005:6 325-337.
• Baryshnikov, Yu. GUEs and queues. Probab. Theory Related Fields 119 (2001), no. 2, 256–274.
• Doumerc, Yan. A note on representations of eigenvalues of classical Gaussian matrices. Séminaire de Probabilités XXXVII, 370–384, Lecture Notes in Math., 1832, Springer, Berlin, 2003.
• Glynn, Peter W.; Whitt, Ward. Departures from many queues in series. Ann. Appl. Probab. 1 (1991), no. 4, 546–572.
• Gravner, Janko; Tracy, Craig A.; Widom, Harold. 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.
• Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
• Johansson, Kurt. Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 (2000), no. 4, 445–456.
• Komlós, J.; Major, P.; Tusnády, G. An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), no. 1, 33–58.
• Krug, J.; Spohn, H. Kinetic roughening of growing surfaces (1992). In C. Godrèche, ed., Solids far from equilibrium Collection Aléa-Saclay: Monographs and Texts in Statistical Physics, 1, pages 479-582. Cambridge University Press, Cambridge.
• Major, Péter. The approximation of partial sums of independent RV's. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 3, 213–220. (54 #3823)
• Martin, James B. Large tandem queueing networks with blocking. Queueing Syst. 41 (2002), no. 1-2, 45–72.
• Martin, James B. Limiting shape for directed percolation models. Ann. Probab. 32 (2004), no. 4, 2908–2937.
• O'Connell, Neil. Random matrices, non-colliding processes and queues. Séminaire de Probabilités, XXXVI, 165–182, Lecture Notes in Math., 1801, Springer, Berlin, 2003.
• O'Connell, Neil; Yor, Marc. A representation for non-colliding random walks. Electron. Comm. Probab. 7 (2002), 1–12 (electronic).
• Tracy, Craig A.; Widom, Harold. Distribution functions for largest eigenvalues and their applications. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 587–596, Higher Ed. Press, Beijing, 2002.