Communications in Mathematical Physics

, Volume 319, Issue 1, pp 231–267 | Cite as

Non-colliding Brownian Motions and the Extended Tacnode Process

  • Kurt JohanssonEmail author


We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.


Brownian Motion Dominate Convergence Theorem Random Matrix Theory Left Half Plane Fredholm Determinant 
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  1. 1.
    Adler M., Moerbeke P.: PDEs for the joint distributions of the Dyson, Airy and Sine processes. Ann. Probab. 33, 1326–1361 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Adler M., Delépine J., Moerbeke P.: Dyson’s nonintersecting brownian motions with a few outliers. Comm. Pure Appl. Math. 62, 334–395 (2010)CrossRefGoogle Scholar
  3. 3.
    Adler M., Ferrari P.L., Moerbeke P.: Airy processes with wanderers and new universality classes. Ann. Probab. 38, 714–769 (2008)CrossRefGoogle Scholar
  4. 4.
    Adler, M., Ferrari, P.L., van Moerbeke, P.: Non-intersecting random walks in the neighborhood of a symmetric tacnode. [math-ph], 2011
  5. 5.
    Adler, M., Johansson, K., van Moerbeke, P.: Double Aztec Diamonds and the Tacnode process. arXiv:1112.5532Google Scholar
  6. 6.
    Adler M., Orantin N., Moerbeke P.: Universality for the Pearcey process. Physica D 239, 924–941 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Adler M., van Moerbeke P., Vanderstichelen D.: Non-interecting Brownian Motions leaving from and going to several points. Phys. D 241(5), 443–460 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Basor E.L., Widom H.: On a Toeplitz determinant identity of Borodin and Okounkov. Int. Eqs. Op. Thy. 37, 397–401 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Borodin A., Duits M.: Limits of determinantal processes near a tacnode. Ann. Inst. H. Poincaré 47(1), 243–258 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Borodin A., Okounkov A.: A Fredholm determinant formula for Toeplitz determinants. Int. Eq. Op. Thy. 37, 386–396 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brézin E., Hikami S.: Extension of level-spacing universality. Phys. Rev. E 56, 264–269 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    Brézin E., Hikami S.: Universal singularity at the closure of a gap. Phys. Rev. E 57, 4140–4149 (1998)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X.: Strong asymptotics for orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Delvaux S., Kuijlaars A.: A phase transition for non-intersecting Brownian motions, and the Painlevé equation. Int. Math. Res. Not. 2009, 3639–3725 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Delvaux S., Kuijlaars A.: A graph based equilibrium problem for the limiting distribution of non-interesecting Brownian motions at low temperature. Constr. Approx. 32, 467–512 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Delvaux S., Kuijlaars A., Zhang L.: Critical behavior of non-intersecting Brownian motions at a tacnode. Commu. Pure Appl. Math. 64, 1305–1383 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Erdös L., Péché S., Ramírez J.A., Schlein B., Yau H.-T.: Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63, 895–925 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Eynard B., Mehta M.L.: Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Ferrari P.L., Spohn H.: Step fluctations for a faceted crystal. J. Stat. Phys. 113, 1–46 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Forrester P.: The Spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Geronimo J.S., Case K.M.: Scattering theory and polynomials orthogonal on the unit circle. J. Math. Phys. 20, 299–310 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Johansson K.: Universality of the local spacing distribution in certain Hermitian Wigner matrices. Commun. Math. Phys. 215, 683–705 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Johansson K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)MathSciNetADSzbMATHGoogle Scholar
  24. 24.
    Johansson K.: Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier 55, 2129–2145 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Karlin S., McGregor L.: Coincidence probabilities. Pacific J. 9, 1141–1164 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Katori M., Tanemura H.: Noncolliding Brownian Motion and Determinantal Processes. J. Stat. Phys. 129, 1233–1277 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Nagao T., Forrester P.J.: Multilevel dynamical correlation functions for Dysons Brownian motion model of random matrices. Phys. Lett. A 247, 42–46 (1998)ADSCrossRefGoogle Scholar
  28. 28.
    Okounkov A., Reshetikhin N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16, 581–603 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Okounkov A., Reshetikhin N.: Random skew plane partitions and the Pearcey process. Commun. Math. Phys. 269, 571–609 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Prähofer M., Spohn H.: Scale invariance of the PNG droplet and the Airy Process. J. Stat. Phys. 108, 1076–1106 (2002)CrossRefGoogle Scholar
  31. 31.
    Sagan, B.E.: The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Second edition. Graduate Texts in Mathematics, 203. New York: Springer-Verlag, 2001Google Scholar
  32. 32.
    Soshnikov A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Tracy C., Widom H.: The Pearcey Process. Commun. Math. Phys. 263, 381–400 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. 34.
    Vanlessen M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. Constr. Approx. 25, 125–175 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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