Limits of determinantal processes near a tacnode
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 243-258.

Nous étudions un processus de Markov sur un système de particules entrelacées. Lorsque le temps t est grand, les particules remplissent un domaine dépendant d'un paramètre ε > 0. Ce domaine possède deux points de rebroussement, dont l'un pointe vers le haut et l'autre vers le bas. À la limite ε ↓ 0, les deux points de rebroussement sont tangents, formant ainsi un tacnode. Le résultat principal de cet article est un calcul du noyau de corrélation locale autour du point tacnodal pendant le régime de transition ε ↓ 0. Nous démontrons aussi que le processus local interpole entre le processus de Pearcey et le processus des mineurs du GUE.

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.

DOI : 10.1214/10-AIHP373
Classification : 82C22, 60K35, 60G55
Mots-clés : determinantal point processes, random growth, GUE minor process, pearcey process 
@article{AIHPB_2011__47_1_243_0,
     author = {Borodin, Alexei and Duits, Maurice},
     title = {Limits of determinantal processes near a tacnode},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {243--258},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     doi = {10.1214/10-AIHP373},
     mrnumber = {2779404},
     zbl = {1208.82039},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/10-AIHP373/}
}
TY  - JOUR
AU  - Borodin, Alexei
AU  - Duits, Maurice
TI  - Limits of determinantal processes near a tacnode
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 243
EP  - 258
VL  - 47
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/10-AIHP373/
DO  - 10.1214/10-AIHP373
LA  - en
ID  - AIHPB_2011__47_1_243_0
ER  - 
%0 Journal Article
%A Borodin, Alexei
%A Duits, Maurice
%T Limits of determinantal processes near a tacnode
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 243-258
%V 47
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/10-AIHP373/
%R 10.1214/10-AIHP373
%G en
%F AIHPB_2011__47_1_243_0
Borodin, Alexei; Duits, Maurice. Limits of determinantal processes near a tacnode. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 243-258. doi : 10.1214/10-AIHP373. http://archive.numdam.org/articles/10.1214/10-AIHP373/

[1] M. Adler, P. Ferrari and P. V. Moerbeke. Airy processes with wanderers and new universality classes. Available at arXiv:0811.1863. | Zbl

[2] A. Aptekarev, P. Bleher and A. Kuijlaars. Large n limit of Gaussian random matrices with external source, part II. Comm. Math. Phys. 259 (2005) 367-389. | MR | Zbl

[3] A. Borodin. Periodic Schur process and cylindric partitions. Duke Math. J. 10 (2007) 1119-1178. | MR | Zbl

[4] A. Borodin. Determinantal point processes. Available at arXiv:0911.1153. | Zbl

[5] A. Borodin and P. Ferrari. Anisotropic growth of random surfaces in 2+1 dimensions. Available at arXiv:0804.3035. | Zbl

[6] A. Borodin and G. Olshanski. Asymptotics of Plancherel-type random partitions. J. Algebra 313 (2007) 40-60. | MR | Zbl

[7] A. Borodin, G. Olshanski and A. Okounkov. Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (2000) 481-515. | MR | Zbl

[8] E. Brezin and S. Hikami. Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E (3) 57 (1998) 7176-7185. | MR

[9] E. Brezin and S. Hikami. Level spacing of random matrices in an external source. Phys. Rev. E (3) 58 (1998) 4140-4149. | MR

[10] P. Ferrari. The universal Airy1 and Airy2 processes in the totally asymmetric simple exclusion process. In Integrable Systems and Random Matrices 321-332. Contemp. Math. 458. Amer. Math. Soc., Providence, RI, 2008. | MR | Zbl

[11] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág. Determinantal processes and independence. Probab. Surv. 3 (2006) 206-229. | MR | Zbl

[12] K. Johansson. Random matrices and determinantal processes. Available at arXiv:math-ph/0510038. | MR

[13] K. Johansson and E. Nordenstam. Eigenvalues of GUE minors. Electron. J. Probab. 11 (2006) 1342-1371. | MR | Zbl

[14] W. König. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005) 385-447. | MR | Zbl

[15] R. Lyons. Determinantal probability measures. Publ. Math. Inst. Hautes Etudes Sci. 98 (2003) 167-212. | Numdam | MR | Zbl

[16] A. Okounkov. Symmetric functions and random partitions. In Symmetric Functions 2001: Surveys of Developments and Perspectives 223-252. NATO Sci. Ser. II Math. Phys. Chem. 74. Kluwer Academic, Dordrecht, 2002. | MR | Zbl

[17] A. Okounkov and N. Reshetikhin. The birth of a random matrix. Moscow Math. J. 6 (2006) 553-566. | MR | Zbl

[18] A. Okounkov and N. Reshetikhin. Random skew plane partitions and the Pearcey process. Comm. Math. Phys. 269 (2007) 571-609. | MR | Zbl

[19] M. Prähofer and H. Spohn. Scale invariance of the PNG Droplet and the Airy Process. J. Stat. Phys. 108 (2002) 1071-1106. | MR | Zbl

[20] A. Soshnikov. Determinantal random point fields. Uspekhi Mat. Nauk 55 (2000) 107-160; translation in: Russian Math. Surveys 55 (2000) 923-975. | MR | Zbl

[21] A. Soshnikov. Determinantal random point fields. In Encyclopedia of Mathematical Physics 2 47-53. J. P. Françoise, G. L. Naber and T. S. Tsun (Eds.). Elsevier, Oxford, 2006. | Zbl

[22] C. Tracy and H. Widom. The Pearcey process. Comm. Math. Phys 263 (2006) 381-400. | MR | Zbl

Cité par Sources :