Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices

Details Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices

2003-03-27

arxiv.org/abs/cond-mat/0303573v2

Phys. Rev. E 68 (2003) 021112

Physics

Condensed Matter

Statistical Mechanics

REVTeX4, 16 pages, 4 figures. v2: some additions and corrections

Scientific paper

10.1103/PhysRevE.68.021112

Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is a motion of radial coordinate of the three-dimensional Brownian motion represented in the spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue-statistics of Gaussian ensembles of Bogoliubov-deGennes Hamiltonians of the mean-field theory of superconductivity, which have the particle-hole symmetry. We report that the time evolution of the present stochastic process is fully characterized by the change of symmetry classes from the type $C$ to the type $C$I in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the non-colliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.

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