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

Makoto Katori·Hideki Tanemura·Taro Nagao·Naoaki Komatsuda
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Abstract

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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Makoto Katori, Hideki Tanemura, Taro Nagao, Naoaki Komatsuda. 2003-06-24. Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices. https://doi.org/10.1103/physreve.68.021112

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