Abstract
Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν > − 1, in which the inhomogeneity is indexed by \(\kappa \in [0, 2(\nu+1))\). We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann–Liouville differintegrals of functions comprising the Bessel functions J ν used in the fractional calculus, where orders of differintegration are determined by ν − κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.
Article PDF
Similar content being viewed by others
References
Altland A., Zirnbauer M.R. (1996) Random matrix theory of a chaotic Andreev quantum dot. Phys. Rev. Lett. 76, 3420–3424
Altland A., Zirnbauer M.R. (1997) Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142–1161
Andrews G.E., Askey R., Roy R. (1999) Special Functions. Cambridge University Press, Cambridge
Borodin A., Rains E.M. (2005) Eynard–Mehta theorem, Schur process, and their pfaffian analogs. J. Stat. Phys. 121, 291–317
Borodin A.N., Salminen P. (2002) Handbook of Brownian Motion—Facts and Formulae, 2nd ed., Birkhäuser, Basel
Borodin A., Soshnikov A. (2003) Janossy densities. I. Determinantal ensembles. J. Stat. Phys. 113, 595–610
Bru M.F. (1989) Diffusions of perturbed principal component analysis. J. Multivar. Anal. 29, 127–136
Caselle M., Magnea U. (2004) Random matrix theory and symmetric spaces. Phys. Rep. 394, 41–156
de Bruijn N.G. (1955) On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133–151
Dyson F.J. (1962) A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198
Dyson F.J. (1962) The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215
Dyson F.J. (1970) Correlation between the eigenvalues of a random matrix. Commun. Math. Phys. 19, 235–250
Ferrari P.L. (2004) Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Commun. Math. Phys. 252, 77–109
Ferrari P.L., Spohn H. (2003) Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1–46
Forrester P.J., Nagao T., Honner G. (1999) Correlations for the orthogonal–unitary and symplectic–unitary transitions at the hard and soft edges. Nucl. Phys. B553[PM]: 601–643
Grabiner D.J. (1999) Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincaré 35, 177–204
Imamura T., Sasamoto T. (2005) Polynuclear growth model with external source and random matrix model with deterministic source. Phys. Rev. E 71, 041606
Ivanov, D.A. Random-matrix ensembles in p-wave vortices. http://arxiv.org/abs/cond-mat/ 0103089 (2001)
Jackson A.D., Sener M.K., Verbaarschot J.J.M. (1996) Finite volume partition functions and Itzykson–Zuber integrals. Phys. Lett. B387, 355–360
Johansson K. (2003) Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329
Karlin S., McGregor J. (1959) Coincidence probabilities. Pac. J. Math. 9, 1141–1164
Katori M., Nagao T., Tanemura H. (2003) Infinite systems of non-colliding Brownian particles. Adv. Stud. Pure Math. 39, 283–306 (2004); http://arxiv.org/abs/math.PR/0301143
Katori M., Tanemura H. (2002) Scaling limit of vicious walks and two–matrix model. Phys. Rev. E 66, 011105
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)
Katori M., Tanemura H. (2003) Noncolliding Brownian motions and Harish–Chandra formula. Elect. Comm. Probab. 8, 112–121
Katori M., Tanemura H. (2004) Symmetry of matrix-valued stochastic process and noncolliding diffusion particle systems. J. Math. Phys. 45, 3058–3085
Katori M., Tanemura H., Nagao T., Komatsuda N. (2003) Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov–deGennes random matrices. Phys. Rev. E 68, 021112
König W., O’Connell N. (2001) Eigenvalues of the Laguerre process as non-colliding squared Bessel processes. Elect. Comm. Probab. 6, 107–114
Mehta, M.L. Matrix Theory. Editions de Physique, Orsay (1989)
Mehta M.L. (2004) Random Matrices, 3rd ed., Elsevier, Amsterdam
Nagao T. (2001) Correlation functions for multi-matrix models and quaternion determinants. Nucl. Phys. B602, 622–637
Nagao T. (2003) Dynamical correlations for vicious random walk with a wall. Nucl. Phys. B658[FS]: 373–396
Nagao T., Forrester P.J. (1999) Quaternion determinant expressions for multilevel dynamical correlation functions of parametric random matrices. Nucl. Phys. B563[PM]: 547–572
Nagao T., Katori M., Tanemura H. (2003) Dynamical correlations among vicious random walkers. Phys. Lett. A 307, 29–35
Okounkov A., Reshetikhin N. (2003) Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581–603
Oldham K.B., Spanier J. (1974) The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic, New York
Osada H. (2004) 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
Podlubny I. (1999) Fractional Differential Equations; an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications. Academic, San Diego
Prähofer M., Spohn H. (2002) Scale invariance of the PNG droplet and the airy process. J. Stat. Phys. 108, 1071–1106
Rains, E.M. Correlation functions for symmetrized increasing subsequences. http://arxiv.org/ abs/math.CO/0006097 (2000)
Revuz D., Yor M. (1998) Continuous Martingales and Brownian Motion. 3rd ed., Springer, Berlin Heidelberg New York
Riordan J. (1979) Combinatorial Identities. R.E. Krieger, New York
Rogers L.C.G., Shi Z. (1993) Interacting Brownian particles and the Wigner law. Probab. Theory Relat. Fields 95, 555–570
Rubin B. (1996) Fractional Integrals and Potentials. Addison–Wesley, Harlow
Sasamoto T., Imamura T. (2004) Fluctuations of the one-dimensional polynuclear growth model in half space. J. Stat. Phys. 115, 749–803
Sener M.K., Verbaarschot J.J.M. (1998) Universality in chiral random matrix theory at β = 1 and β = 4. Phys. Rev. Lett. 81, 248–251
Soshnikov A. (2003) Janossy densities. II. Pfaffian ensemble. J. Stat. Phys. 113, 611–622
Soshnikov A. (2004) Janossy densities of coupled random matrices. Commun. Math. Phys. 251, 447–471
Spohn, H. Dyson’s model of interacting Brownian motions at arbitrary coupling strength (preprint)
Tracy C.A., Widom H. (2004) Differential equations for Dyson processes. Commun. Math. Phys. 252, 7–41
Verbaarschot J. (1994) The spectrum of the Dirac operator near zero virtuality for N c = 2 and chiral random matrix theory. Nucl. Phys. B426[FS]: 559–574
Verbaarschot J.J.M., Zahed I. (1993) Spectral density of the QCD Dirac operator near zero virtuality. Phys. Rev. Lett. 70, 3852–3855
Yor M. (1992) Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkhäuser, Basel
Zirnbauer M.R. (1996) Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986–5018
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Katori, M., Tanemura, H. Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals. Probab. Theory Relat. Fields 138, 113–156 (2007). https://doi.org/10.1007/s00440-006-0015-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-006-0015-4
Keywords
- Noncolliding generalized meanders
- Bessel processes
- Random matrix theory
- Fredholm Pfaffian and determinant
- Riemann–Liouville differintegrals