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

TL;DR

This paper links the evolution of one-dimensional vicious walkers with wall restrictions to noncolliding stochastic processes, revealing connections to random matrix theory and symmetry class transitions relevant to superconductivity.

Contribution

It establishes a novel equivalence between vicious walkers with a wall and Brownian meanders, and connects their spatial distributions to eigenvalue statistics of Bogoliubov-deGennes Hamiltonians, highlighting symmetry class changes.

Findings

Vicious walkers with a wall are described by noncolliding Brownian meanders.

Spatial distributions relate to eigenvalue statistics of Bogoliubov-deGennes matrices.

Time evolution involves symmetry class transitions from type C to C I.

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.