Limits of determinantal processes near a tacnode

Alexei Borodin; Maurice Duits

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 1, page 243-258
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Borodin, Alexei, and Duits, Maurice. "Limits of determinantal processes near a tacnode." Annales de l'I.H.P. Probabilités et statistiques 47.1 (2011): 243-258. <http://eudml.org/doc/243428>.

@article{Borodin2011,
abstract = {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 &gt; 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.},
author = {Borodin, Alexei, Duits, Maurice},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {determinantal point processes; random growth; GUE minor process; pearcey process; Pearcey process},
language = {eng},
number = {1},
pages = {243-258},
publisher = {Gauthier-Villars},
title = {Limits of determinantal processes near a tacnode},
url = {http://eudml.org/doc/243428},
volume = {47},
year = {2011},
}

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
PB - Gauthier-Villars
VL - 47
IS - 1
SP - 243
EP - 258
AB - 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 &gt; 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.
LA - eng
KW - determinantal point processes; random growth; GUE minor process; pearcey process; Pearcey process
UR - http://eudml.org/doc/243428
ER -

References

top
  1. [1] M. Adler, P. Ferrari and P. V. Moerbeke. Airy processes with wanderers and new universality classes. Available at arXiv:0811.1863. Zbl1200.60069
  2. [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. Zbl1129.82014MR2172687
  3. [3] A. Borodin. Periodic Schur process and cylindric partitions. Duke Math. J. 10 (2007) 1119–1178. Zbl1131.22003MR2362241
  4. [4] A. Borodin. Determinantal point processes. Available at arXiv:0911.1153. 
  5. [5] A. Borodin and P. Ferrari. Anisotropic growth of random surfaces in 2+1 dimensions. Available at arXiv:0804.3035. Zbl1303.82015
  6. [6] A. Borodin and G. Olshanski. Asymptotics of Plancherel-type random partitions. J. Algebra 313 (2007) 40–60. Zbl1117.60051MR2326137
  7. [7] A. Borodin, G. Olshanski and A. Okounkov. Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (2000) 481–515. Zbl0938.05061MR1758751
  8. [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. MR1618958
  9. [9] E. Brezin and S. Hikami. Level spacing of random matrices in an external source. Phys. Rev. E (3) 58 (1998) 4140–4149. MR1662382
  10. [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. Zbl1145.82332MR2411915
  11. [11] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág. Determinantal processes and independence. Probab. Surv. 3 (2006) 206–229. Zbl1189.60101MR2216966
  12. [12] K. Johansson. Random matrices and determinantal processes. Available at arXiv:math-ph/0510038. MR2581882
  13. [13] K. Johansson and E. Nordenstam. Eigenvalues of GUE minors. Electron. J. Probab. 11 (2006) 1342–1371. Zbl1127.60047MR2268547
  14. [14] W. König. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005) 385–447. Zbl1189.60024MR2203677
  15. [15] R. Lyons. Determinantal probability measures. Publ. Math. Inst. Hautes Etudes Sci. 98 (2003) 167–212. Zbl1055.60003MR2031202
  16. [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. Zbl1017.05103MR2059364
  17. [17] A. Okounkov and N. Reshetikhin. The birth of a random matrix. Moscow Math. J. 6 (2006) 553–566. Zbl1130.15014MR2274865
  18. [18] A. Okounkov and N. Reshetikhin. Random skew plane partitions and the Pearcey process. Comm. Math. Phys. 269 (2007) 571–609. Zbl1115.60011MR2276355
  19. [19] M. Prähofer and H. Spohn. Scale invariance of the PNG Droplet and the Airy Process. J. Stat. Phys. 108 (2002) 1071–1106. Zbl1025.82010MR1933446
  20. [20] A. Soshnikov. Determinantal random point fields. Uspekhi Mat. Nauk 55 (2000) 107–160; translation in: Russian Math. Surveys 55 (2000) 923–975. Zbl0991.60038MR1799012
  21. [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. Zbl0991.60038
  22. [22] C. Tracy and H. Widom. The Pearcey process. Comm. Math. Phys 263 (2006) 381–400. Zbl1129.82031MR2207649

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.