Noncolliding Brownian motion and determinantal processes

Journal of Statistical Physics

Volumn 129, Issue 5-6, 2007, Pages 1233-1277

  Katori, Makoto ^a   Tanemura, Hideki ^b  

^a CHUO UNIVERSITY   (Japan)

^b CHIBA UNIVERSITY   (Japan)

Author keywords

Determinantal processes; Karlin McGregor formula; Matrix kernels; Multitime correlation functions; Noncolliding Brownian motion; Random matrix theory; Spectral projections

Indexed keywords

EID: 36448991470     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-007-9421-y     Document Type: Article

References (99)

1. Adler, M., van Moerbeke, P.: The spectrum of coupled random matrices. Ann. Math. 149, 921-976 (1999)
2. Adler, M., van Moerbeke, P.: PDF's for the joint distributions of the Dyson, Airy and Sine processes. Ann. Probab. 33, 1326-1361 (2005)
3. Altland, A., Zirnbauer, M.R.: Random matrix theory of a chaotic Andreev quantum dot. Phys. Rev. Lett. 76, 3420-3424 (1996)
4. Altland, A., Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142-1161 (1997)
5. Aptekarev, A.I., Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, part II. Commun. Math. Phys. 259, 367-389 (2005)
6. Arcioni, G., de Haro, S., Gao, P.: Diffusion model for SU(N) QCD screening. Phys. Rev. D 73, 074508 (2006)
7. Baik, J., Borodin, A., Deift, P., Suidan, T.: A model for the bus system in Cuernavaca (Mexico). J. Phys. A: Math. Gen. 39, 8965-8975 (2006)
8. Balantekin, A.B.: Character expansions, Itzykson-Zuber integrals, and the QCD partition function. Phys. Rev. D 62, 0850171 (2000)
9. Balantekin, A.B.: Character expansions for the orthogonal and symplectic groups. J. Math. Phys. 43, 604-620 (2002)
10. Bateman, H.: Higher Transcendental Functions (A. Erdélyi ed.), vol. 2. McGraw-Hill, New York (1953)
11. Baym, G.: Lectures on Quantum Mechanics. Benjamin, Elmsford (1969)
12. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Willey, New York (1999)
13. Bogoliubov, N.M.: XX0 Heisenberg chain and random walks. J. Math. Sci. 138, 5636-5643 (2006)
14. Borodin, A., Olshanski, G.: Markov processes on partitions. Probab. Theory Relat. Fields 135, 84-152 (2006)
15. Borodin, A., Olshanski, G.: Asymptotics of Plancherel-type random partitions. J. Algebra 313, 40-60 (2007)
16. Borodin, A., Olshanski, G.: Infinite-dimensional diffusions as limits of random walks on partitions. arXiv:math.PR/07061034 (2007)
17. Borodin, A., Rains, E.M.: Eynard-Mehta theorem, Schur process, and their pfaffian analogs. J. Stat. Phys. 121, 291-317 (2005)
18. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae, 2nd edn. Birkhäuser, Basel (2002)
19. Borodin, A., Soshnikov, A.: Janossy densities. I. Determinantal ensembles. J. Stat. Phys. 113, 595-610 (2003)
20. Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 57, 4140-4149 (1998)
21. Brézin, E., Hikami, S.: Level spacing of random matrices in an external sources. Phys. Rev. E 58, 7176-7185 (1998)
22. Bleher, P.M., Kuijlaars, A.B.J.: Random matrices with external sources and multiple orthogonal polynomials. Int. Math. Res. Not. 2004, 109-129 (2004)
23. Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external sources, part I. Commun. Math. Phys. 252, 43-76 (2004)
24. Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external sources, part III: double scaling limit. Commun. Math. Phys. 270, 481-517 (2007)
25. de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133-151 (1955)
26. de Gennes, P.-G.: Soluble model for fibrous structures with steric constraints. J. Chem. Phys. 48, 2257-2259 (1968)
27. de Haro, S.: Chern-Simons theory, 2d Yang-Mills, and Lie algebra wanders. Nucl. Phys. B 730, 312-351 (2005)
28. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, London (1958)
29. Doob, J.L.: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York (1984)
30. Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191-1198 (1962)
31. Dyson, F.J.: Correlation between the eigenvalues of a random matrix. Commun. Math. Phys. 19, 235-250 (1970)
32. Eichelsbacher, P., König, W.: Ordered random walks. arXiv:math.PR/0610850 (2006)
33. Essam, J.W., Guttmann, A.J.: Vicious walkers and directed polymer networks in general dimensions. Phys. Rev. E 52, 5849-5862 (1995)
34. Ferrari, P.L.: Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Commun. Math. Phys. 252, 77-109 (2004)
35. Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1-46 (2003)
36. Fisher, M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667-729 (1984)
37. Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B 553(3), 601-643 (1999)
38. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin (1994)
39. Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, Cambridge (1997)
40. Gessel, I., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58, 300-321 (1985)
41. Grabiner, D.J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 35, 177-204 (1999)
42. Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes, 2nd edn. Clarendon, Oxford (1992)
43. Harnad, J., Orlov, A.Y.: Fermionic construction of tau functions and random processes. Physica D (2007, in press); arXiv:math-ph/07041157 (2007)
44. Imamura, T., Sasamoto, T.: Polynuclear growth model with external source and random matrix model with deterministic source. Phys. Rev. E 71, 041606 (2005)
45. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123, 225-280 (2002)
46. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277-329 (2003)
47. Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (1997)
48. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)
49. Karlin, S., McGregor, J.: Coincidence probabilities. Pac. J. Math. 9, 1141-1164 (1959)
50. Katori, M.: Percolation transitions and wetting transitions in stochastic models. Braz. J. Phys. 30, 83-96 (2000)
51. Katori, M., Nagao, T., Tanemura, H.: Infinite systems of non-colliding Brownian particles. In: Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math., vol. 39, pp. 283-306. Mathematical Society of Japan, Tokyo (2004); arXiv:math.PR/0301143
52. Katori, M., Tanemura, H.: Scaling limit of vicious walks and two-matrix model. Phys. Rev. E 66, 011105 (2002)
53. Katori, M., Tanemura, H.: Functional central limit theorems for vicious walkers. Stoch. Stoch. Rep. 75, 369-390 (2003); arXiv:math.PR/0203286
54. Katori, M., Tanemura, H.: Noncolliding Brownian motions and Harish-Chandra formula. Elect. Comm. Probab. 8, 112-121 (2003)
55. Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45, 3058-3085 (2004)
56. Katori, M., Tanemura, H.: Nonintersecting paths, Noncolliding diffusion processes and representation theory. RIMS Kokyuroku 1438, 83-102 (2005); arXiv:math.PR/0501218
57. Katori, M., Tanemura, H.: Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals. Probab. Theory Relat. Fields 138, 113-156 (2007)
58. Katori, M., Tanemura, H., Nagao, T., Komatsuda, N.: Vicious walk with a wall, noncolliding meanders, chiral and Bogoliubov-de Gennes random matrices. Phys. Rev. E 68, 0211121-16 (2003)
59. König, W., O'Connell, N.: Eigenvalues of the Laguerre process as non-colliding squared Bessel process. Electron. Commun. Probab. 6, 107-114 (2001)
60. Krattenthaler, C.: Watermelon configurations with wall interaction: exact and asymptotic results. J. Phys. Conf. Ser. 42, 179-212 (2006)
61. Krattenthaler, C., Guttmann, A.J., Viennot, X.G.: Vicious walkers, friendly walkers and Young tableaux: II. With a wall. J. Phys. A: Math. Phys. 33, 8835-8866 (2000)
62. Lindström, B.: On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5, 85-90 (1973)
63. Macdonald, I.G.: Some conjectures for root systems. SIAM J. Math. Anal. 13, 988-1007 (1982)
64. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)
65. Mehta, M.L.: Matrix Theory. Editions de Physique, Orsay (1989)
66. Mehta, M.L.: Random Matrices, 3rd edn. Elsevier, Amsterdam (2004)
67. Mehta, M.L., Pandey, A.: On some Gaussian ensemble of Hermitian matrices. J. Phys. A: Math. Gen. 16, 2655-2684 (1983)
68. Nagao, T.: Correlation functions for multi-matrix models and quaternion determinants. Nucl. Phys. B 602, 622-637 (2001)
69. Nagao, T.: Dynamical correlations for vicious random walk with a wall. Nucl. Phys. B 658(3), 373-396 (2003)
70. Nagao, T., Forrester, P.J.: Multilevel dynamical correlation function for Dyson's Brownian motion model of random matrices. Phys. Lett. A 247, 42-46 (1998)
71. Nagao, T., Forrester, P.J.: Quaternion determinant expressions for multilevel dynamical correlation functions of parametric random matrices. Nucl. Phys. B 563(3), 547-572 (1999)
72. Nagao, T., Katori, M., Tanemura, H.: Dynamical correlations among vicious random walkers. Phys. Lett. A 307, 29-35 (2003)
73. 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)
74. Okounkov, A., Reshetikhin, N.: Random skew plane partitions and the Pearcey process. Commun. Math. Phys. 269, 571-609 (2007)
75. Osada, H.: Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions. Commun. Math. Phys. 176, 117-131 (1996)
76. 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. In: Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math., vol. 39, pp. 325-343. Mathematical Society of Japan, Tokyo (2004)
77. Pandey, A., Mehta, M.L.: Gaussian ensembles of random Hermitian intermediate between orthogonal and unitary ones. Commun. Math. Phys. 87, 449-468 (1983)
78. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071-1106 (2002)
79. Rains, E.M.: Correlation functions for symmetrized increasing subsequences. arXiv:math.CO/0006097 (2000)
80. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York (1998)
81. Rogers, L.C.G., Shi, Z.: Interacting Brownian particles and the Wigner law. Probab. Theory Relat. Fields 95, 555-570 (1993)
82. Sasamoto, T., Imamura, T.: Fluctuations of the one-dimensional polynuclear growth model in half space. J. Stat. Phys. 115, 749-803 (2004)
83. Selberg, A.: Bemerkninger om et multiplet integral. Nor. Mat. Tidsskr. 26, 71-78 (1944)
84. Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point process. J. Funct. Anal. 205, 414-436 (2003)
85. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923-975 (2000)
86. Soshnikov, A.: Janossy densities. II. Pfaffian ensemble. J. Stat. Phys. 113, 611-622 (2003)
87. Soshnikov, A.: Janossy densities of coupled random matrices. Commun. Math. Phys. 251, 447-471 (2004)
88. Spohn, H.: Interacting Brownian particles: a study of Dyson's model. In: Papanicolaou, G. (ed.) Hydrodynamic Behavior and Interacting Particle Systems. IMA Volumes in Mathematics and its Applications, vol. 9, pp. 151-179. Springer, Berlin (1987)
89. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
90. Stembridge, J.R.: Nonintersecting paths, pfaffians, and the plane partitions. Adv. Math. 83, 96-131 (1990)
91. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)
92. Tanemura, H., Yoshida, N.: Localization transition of d-friendly walkers. Probab. Theory Relat. Fields 125, 593-608 (2003)
93. Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, 2nd edn. Clarendon, Oxford (1962)
94. Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151-174 (1994)
95. Tracy, C.A., Widom, H.: A system of differential equations for the Airy process. Electron. Commun. Probab. 8, 93-98 (2003)
96. Tracy, C.A., Widom, H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7-41 (2004)
97. Tracy, C.A., Widom, H.: The Pearcey process. Commun. Math. Phys. 263, 381-400 (2006)
98. Tracy, C.A., Widom, H.: Nonintersecting Brownian excursions. Ann. Appl. Probab. 17, 953-979 (2007)
99. Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986-5018 (1996)