Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals

Probability Theory and Related Fields

Volumn 138, Issue 1-2, 2007, Pages 113-156

  Katori, Makoto ^a   Tanemura, Hideki ^b  

^a CHUO UNIVERSITY   (Japan)

^b CHIBA UNIVERSITY   (Japan)

Author keywords

Bessel processes; Fredholm Pfaffian and determinant; Noncolliding generalized meanders; Random matrix theory; Riemann Liouville differintegrals

Indexed keywords

EID: 33847680999     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-006-0015-4     Document Type: Article

References (54)

1. Altland, A., Zirnbauer, M.R.: Random matrix theory of a chaotic Andreev quantum dot. Phys. Rev. Lett. 76, 3420-3424 (1996)
2. Altland, A., Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142-1161 (1997)
3. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
4. Borodin, A., Rains, E.M.: Eynard-Mehta theorem, Schur process, and their pfaffian analogs. J. Stat. Phys. 121, 291-317 (2005)
5. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae, 2nd ed., Birkhäuser, Basel (2002)
6. Borodin. A., Soshnikov, A.: Janossy densities. I. Determinantal ensembles. J. Stat. Phys. 113, 595-610 (2003)
7. Bru, M.F.: Diffusions of perturbed principal component analysis. J. Multivar. Anal. 29, 127-136 (1989)
8. Caselle, M., Magnea, U.: Random matrix theory and symmetric spaces. Phys. Rep. 394, 41-156 (2004)
9. de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133-151 (1955)
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. Dyson, F.J.: Correlation between the eigenvalues of a random matrix. Commun. Math. Phys. 19, 235-250 (1970)
13. Ferrari, P.L.: Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Commun. Math. Phys. 252, 77-109 (2004)
14. Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1-46 (2003)
15. Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B553[PM], 601-643 (1999)
16. Grabiner, D.J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincaré 35, 177-204 (1999)
17. Imamura, T., Sasamoto, T.: Polynuclear growth model with external source and random matrix model with deterministic source. Phys. Rev. E 71, 041606 (2005)
18. Ivanov, D.A.: Random-matrix ensembles in p-wave vortices. http://arxiv.org/abs/cond-mat/0103089 (2001)
19. Jackson, A.D., Sener, M.K., Verbaarschot, J.J.M.: Finite volume partition functions and Itzykson. Zuber integrals. Phys. Lett. B387, 355-360 (1996)
20. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277-329 (2003)
21. Karlin. S., McGregor, J.: Coincidence probabilities. Pac. J. Math. 9, 1141-1164 (1959)
22. Katori, M., Nagao, T., Tanemura, H.: Infinite systems of non-colliding Brownian particles. Adv. Stud. Pure Math. 39, 283-306 (2004); http://arxiv.org/abs/math.PR/0301143 (2003)
23. Katori, M., Tanemura, H.: Scaling limit of vicious walks and two.matrix model. Phys. Rev. E 66, 011105 (2002)
24. Katori, M., Tanemura, H.: Functional central limit theorems for vicious walkers. Stoch. Stoch. Rep. 75, 369-390 (2003); http://arxiv.org/abs/ math.PR/0203286 (2002)
25. Katori, M., Tanemura, H.: Noncolliding Brownian motions and Harish-Chandra formula. Elect. Comm. Probab. 8, 112-121 (2003)
26. Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic process and noncolliding diffusion particle systems. J. Math. Phys. 45, 3058-3085 (2004)
27. Katori, M., Tanemura, H., Nagao, T., Komatsuda, N.: Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices. Phys. Rev. E 68, 021112 (2003)
28. König,W., O'Connell, N.: Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Elect. Comm. Probab. 6, 107-114 (2001)
29. Mehta, M.L.: Matrix Theory. Editions de Physique, Orsay (1989)
30. Mehta, M.L.: Random Matrices, 3rd ed., Elsevier, Amsterdam (2004)
31. Nagao, T.: Correlation functions for multi-matrix models and quaternion determinants. Nucl. Phys. B602, 622-637 (2001)
32. Nagao, T.: Dynamical correlations for vicious random walk with a wall. Nucl. Phys. B658[FS], 373-396 (2003)
33. Nagao, T., Forrester, P.J.: Quaternion determinant expressions for multilevel dynamical correlation functions of parametric random matrices. Nucl. Phys. B563[PM], 547-572 (1999)
34. Nagao, T., Katori, M., Tanemura, H.: Dynamical correlations among vicious random walkers. Phys. Lett. A 307, 29-35 (2003)
35. Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581-603 (2003)
36. Oldham, K.B., Spanier, J.: The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic, New York (1974)
37. Osada, H.: Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields. Adv. Stud. Pure Math. 39, 325-343 (2004)
38. Podlubny, I.: Fractional Differential Equations; an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications. Academic, San Diego (1999)
39. Prähofer,M., Spohn, H.: Scale invariance of the PNG droplet and the airy process. J. Stat. Phys. 108, 1071-1106 (2002)
40. Rains, E.M.: Correlation functions for symmetrized increasing subsequences. http://arxiv.org/abs/math.CO/0006097 (2000)
41. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. 3rd ed., Springer, Berlin Heidelberg New York (1998)
42. Riordan, J.: Combinatorial Identities. R.E. Krieger, New York (1979)
43. Rogers, L.C.G., Shi, Z.: Interacting Brownian particles and the Wigner law. Probab. Theory Relat. Fields 95, 555-570 (1993)
44. Rubin, B.: Fractional Integrals and Potentials. Addison-Wesley, Harlow (1996)
45. Sasamoto, T., Imamura, T.: Fluctuations of the one-dimensional polynuclear growth model in half space. J. Stat. Phys. 115, 749-803 (2004)
46. Sener, M.K., Verbaarschot, J.J.M.: Universality in chiral random matrix theory at β = 1 and β = 4. Phys. Rev. Lett. 81, 248-251 (1998)
47. Soshnikov, A.: Janossy densities. II. Pfaffian ensemble. J. Stat. Phys. 113, 611-622 (2003)
48. Soshnikov, A.: Janossy densities of coupled random matrices. Commun. Math. Phys. 251, 447-471 (2004)
49. Spohn, H.: Dyson's model of interacting Brownian motions at arbitrary coupling strength (preprint)
50. Tracy, C.A., Widom, H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7-41 (2004)
51. c = 2 and chiral random matrix theory. Nucl. Phys. B426[FS], 559-574 (1994)
52. Verbaarschot, J.J.M., Zahed, I.: Spectral density of the QCD Dirac operator near zero virtuality. Phys. Rev. Lett. 70, 3852-3855 (1993)
53. Yor, M.: Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkhäuser, Basel (1992)
54. Zirnbauer, M.R.:Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986-5018 (1996)