Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

Journal of Mathematical Physics

Volumn 45, Issue 8, 2004, Pages 3058-3085

  Katori, Makoto ^a   Tanemura, Hideki ^b  

^a CHUO UNIVERSITY   (Japan)

^b CHIBA UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4544227951     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1765215     Document Type: Article

References (56)

1. Altland, A. and Zirnbauer, M. R., "Random matrix theory of a chaotic Andreev quantum dot," Phys. Rev. Lett. 76, 3420-3423 (1996).
2. Altland, A. and Zirnbauer, M. R., "Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structure," Phys. Rev. B 55, 1142-1161 (1997).
3. Balantekin, A. B., "Character expansions, Itzykson-Zuber integrals, and the QCD partition function," Phys. Rev. D 62, 085017 (2000).
4. Balantekin, A. B., "Character expansions for the orthogonal and symplectic groups," J. Math. Phys. 43, 604-620 (2002).
5. Borodin, A. N. and Salminen, P., Handbook of Brownian Motion: Facts and Formulas, 2nd ed. (Birkhäuser, Basel, 2002).
6. Bru, M. F., "Diffusions of perturbed principal component analysis," J. Multivariate Anal. 29, 127-136 (1989).
7. Bru, M. F., "Wishart processes," J. Theor. Probab. 3, 725-751 (1991).
8. de Gennes, P.-G., "Soluble model for fibrous structures with steric constraints," J. Chem. Phys. 48, 2257-2259 (1968).
9. Doob, J. L., Classical Potential Theory and its Probabilistic Counterpart (Springer, New York, 1984).
10. Dyson, F. J., "A Brownian-motion model for the eigenvalues of a random matrix," J. Math. Phys. 3, 1191-1198 (1962).
11. Dyson, F. J., "The threefold way: Algebraic structure of symmetry groups and ensembles in quantum mechanics," J. Math. Phys. 3, 1199-1215 (1962).
12. Edelman, A., "The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law," J. Multivariate Anal. 60, 203-232 (1997).
13. Efetov, K., Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997).
14. Essam, J. W. and Guttmann, A. J., "Vicious walkers and directed polymer networks in general dimensions," Phys. Rev. E 52, 5849-5862 (1995).
15. Fisher, M. E., "Walks, walls, wetting, and melting," J. Stat. Phys. 34, 667-729 (1984).
16. Forrester, P. J., Nagao, T., and Honner, G., "Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges," Nucl. Phys. B 553[PM], 601-643 (1999).
17. Fulton, W., Young Tableaux with Applications to Representation Theory and Geometry (Cambridge University Press, Cambridge, 1997).
18. Fulton, W. and Harris, J., Representation Theory, A First Course (Springer, New York, 1991).
19. Gilmore, R., Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).
20. Ginibre, J., "Statistical ensembles of complex, quaternion, and real matrices," J. Math. Phys. 6, 440-449 (1965).
21. Grabiner, D. J., "Brownian motion in a Weyl chamber, non-colliding particles, and random matrices," Ann. I.H.P. Probab. Stat. 35, 177-204 (1999).
22. Harish-Chandra, "Differential operators on a semisimple Lie algebra," Am. J. Math. 79, 87-120 (1957).
23. Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978).
24. Hua, L., On the Theory of Functions of Several Complex Variables. I, edited by L. Ebner and A. Koráni (American Mathematical Society, Providence, 1963).
25. Imhof, J. P., "Density factorizations for Brownian motion, meander and the three-dimensional Bessel processes, and applications," J. Appl. Probab. 21, 500-510 (1984).
26. Itzykson, C. and Zuber, J.-B., "The planar approximation. II," J. Math. Phys. 21, 411-421 (1980).
27. Jackson, A. D., Sener, M. K., and Verbaarschot, J. J. M., "Finite volume partition functions and Itzykson-Zuber integrals," Phys. Lett. B 387, 355-360 (1996).
28. Karlin, S. and McGregor, J., "Coincidence properties of birth and death processes," Pac. J. Math. 9, 1109-1140 (1959).
29. Karlin, S. and McGregor, J., "Coincidence probabilities," Pac. J. Math. 9, 1141-1164 (1959).
30. Katori, M., Nagao, T., and Tanemura, H., "Infinite systems of noncolliding Brownian particles," Adv. Stud. in Pure Math. 39 "Stochastic Analysis on Large Scale Interacting Systems," (Mathematical Society of Japan, Tokyo, 2004), pp. 283-306; math.PR/0301143.
31. Katori, M. and Tanemura, H., "Scaling limit of vicious walks and two-matrix model," Phys. Rev. E 66, 011105 (2002).
32. Katori, M. and Tanemura, H., "Functional central limit theorems for vicious walkers," Stoch. Stoch. Rep. 75, 369-390 (2003); math.PR/0203286.
33. Katori, M. and Tanemura, H., "Noncolliding Brownian motions and Harish-Chandra formula," Elect. Comm. Probab. 8, 112-121 (2003).
34. Katori, M. and Tanemura, H. (in preparation).
35. Katori, M., Tanemura, H., Nagao, T., and Komatsuda, N., "Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-de Gennes random matrices," Phys. Rev. E 68, 021112 (2003).
36. König, W. and O'Connell, N., "Eigenvalues of the Laguerre process as noncolliding squared Bessel process," Elect. Comm. Probab. 6, 107-114 (2001).
37. Krattenthaler, C., Guttmann, A. J., and Viennot, X. G., "Vicious walkers, friendly walkers and Young tableaux: II. With a wall," J. Phys. A 33, 8835-8866 (2000).
38. Macdonald, I. G., "Some conjectures for root systems," SIAM J. Math. Anal. 13, 988-1007 (1982).
39. Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd ed. (Oxford University Press, Oxford, 1995).
40. Mehta, M. L., Random Matrices, 2nd ed. (Academic, London, 1991).
41. Mehta, M. L. and Pandey, A., "On some Gaussian ensemble of Hermitian matrices," J. Phys. A 16, 2655-2684 (1983).
42. Nagao, T., "Correlation functions for multi-matrix models and quaternion determinants," Nucl. Phys. B 602, 622-637 (2001).
43. Nagao, T., "Dynamical correlations for vicious random walk with a wall," Nucl. Phys. B 658[FS], 373-396 (2003).
44. Nagao, T. and Forrester, P. J., "Quaternion determinant expressions for multilevel dynamical correlation functions of parametric random matrices," Nucl. Phys. B 563[PM], 547-572 (1999).
45. Nagao, T., Katori, M., and Tanemura, H., "Dynamical correlations among vicious random walkers," Phys. Lett. A 307, 29-35 (2003).
46. Norris, J. R., Rogers, L. C. G., and Williams, D., "Brownian motions of ellipsoids," Trans. Am. Math. Soc. 294, 757-765 (1986).
47. Pandey, A. and Mehta, M. L., "Gaussian ensembles of random Hermitian matrices intermediate between orthogonal and unitary ones," Commun. Math. Phys. 87, 449-468 (1983).
48. Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, 3rd ed. (Springer, New York, 1998).
49. Sasamoto, T. and Imamura, T., "Fluctuations of the one-dimensional polynuclear growth model in a half space," J. Stat. Phys. 115, 749-803 (2004).
50. Selberg, A., "Bemerkninger om et multiplet integral," Norsk Matematisk Tidsskrift 26, 71-78 (1944).
51. Sener, M. K. and Verbaarschot, J. J. M., "Universality in chiral random matrix theory at β=1 and β=4," Phys. Rev. Lett. 81, 248-251 (1998).
52. Stanley, R. P., Enumerative Combinatorics (Cambridge University Press, Cambridge, 1999), Vol. 2.
53. c = 2 and chiral random matrix theory," Nucl. Phys. B 426[FS], 559-574 (1994).
54. Verbaarschot, J. J. M. and Zahed, I., "Spectral density of the QCD Dirac operator near zero virtuality," Phys. Rev. Lett. 70, 3852-3855 (1993).
55. Yor, M., Some Aspects of Brownian Motion, Part I: Some Special Functionals (Birkhäuser, Basel, 1992).
56. Zirnbauer, M. R., "Riemannian symmetric superspaces and their origin in random-matrix theory," J. Math. Phys. 37. 4986-5018 (1996).