Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 8 (2003) > Paper 13 open journal systems 


Noncolliding Brownian motions and Harish-Chandra formula

Makoto Katori, Chuo university
Hideki Tanemura, Chiba university


Abstract
We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a finite time interval $(0,T]$. This is a temporally inhomogeneous diffusion process whose transition probability density depends on a value of $T$, and in the limit $T to infty$ it converges to a temporally homogeneous diffusion process called Dyson's model of Brownian motions. It is known that the distribution of particle positions in Dyson's model coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brownian motions. In the present paper we construct such a Hermitian matrix-valued process, whose entries are sums of Brownian motions and Brownian bridges given independently of each other, that its eigenvalues are identically distributed with the particle positions of our temporally inhomogeneous system of noncolliding Brownian motions. As a corollary of this identification we derive the Harish-Chandra formula for an integral over the unitary group.


Full text: PDF

Pages: 112-121

Published on: September 23, 2003
Bibliography
  1. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142-1161 (1997). Math. Review number not available.
  2. M. F. Bru, Diffusions of perturbed principal component analysis, J. Maltivated Anal. 29, 127-136 (1989). MR 90k:62123
  3. M. F. Bru, Wishart process, J. Theoret. Probab. 3, 725-751 (1991). MR 93b:60176
  4. J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, 1984. MR 85k:31001
  5. F. J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3, 1191-1198 (1962). MR 26 #5904
  6. W. Fulton and J. Harris, Representation Theory, Springer, New York 1991. MR 93a:20069
  7. D. J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices, Ann. Inst. Henri Poincar'e 35, 177-204 (1999). MR 2000i:60091
  8. Harish-Chandra, Differential operators on a semisimple Lie algebra, Am. J. Math. 79, 87-120 (1957). MR 18,809d
  9. J. P. Imhof, Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications, J. Appl. Prob. 21, 500-510 (1984). MR 85j:60152
  10. C. Itzykson and J.-B. Zuber, The planar approximation. II, J. Math. Phys. 21, 411-421 (1980). MR 81a:81068
  11. S. Karlin and L. McGregor, Coincidence properties of birth and death processes, Pacific J. 9, 1109-1140 (1959). MR 22 #5071
  12. S. Karlin and L. McGregor, Coincidence probabilities, Pacific J. 9, 1141-1164 (1959). MR 22 #5072
  13. M. Katori, T. Nagao and H. Tanemura, Infinite systems of non-colliding Brownian particles, to be published in Adv. Stud. Pure Math. ``Stochastic Analysis on Large Scale Interacting Systems", Mathematical Society of Japan. (2003). Math. Review number not available.
  14. M. Katori and H. Tanemura, Scaling limit of vicious walks and two-matrix model, Phys. Rev. E 66, 011105 (2002). Math. Review number not available.
  15. M. Katori and H. Tanemura, Functional central limit theorems for vicious walkers to appear in Stoch. Stoch. Rep. Math. Review number not available.
  16. M. Katori and H. Tanemura, in preparation. Math. Review number not available.
  17. M. Katori, H. Tanemura, T. Nagao and N. Komatsuda, Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices, Phys. Rev. E 68, 021112 (2003). Math. Review number not available.
  18. W. K"{o}nig, and N. O'Connell, Eigenvalues of the Laguerre process as non-colliding squared Bessel processes, Elect. Comm. in Probab. 6, 107-114 (2001). MR 2002j:15025
  19. M. L. Mehta, Random Matrices, Academic Press, London 1991 (2nd ed.). MR 92f:82002
  20. M. L. Mehta and A. Pandey, On some Gaussian ensemble of Hermitian matrices, J. Phys. A: Math. Gen. 16, 2655-2684 (1983). MR 85h:81082
  21. T. Nagao, M. Katori and H. Tanemura, Dynamical correlations among vicious random walkers, Phys. Lett. A 307, 29-35 (2003). MR 1 969 523
  22. A. Pandey and M. L. Mehta, Gaussian ensembles of random Hermitian matrices intermediate between orthogonal and unitary ones, Commun. Math. Phys. 87, 449-468 (1983). MR 84e:81085
  23. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1998 (3rd ed.). MR 2000h:60050
  24. M. Yor, Some Aspects of Brownian Motion, Part I: Some Special Functionals, Birkh"auser, Basel 1992. MR 93i:60155
  25. M. R. Zirnbauer, Riemannian symmetric superspaces and their origin in random-matrix theory, J. Math. Phys. 37, 4986-5018 (1996). MR 97m:58012
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X