"Vicious walk with a wall, non-colliding meanders, chiral and Bogoliubov-deGennes random matrices
",
( with Makoto Katori, Taro Nagao and Naoaki Komatsuda)
Phys. Rev. E68(2003)
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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 non-colliding 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.
