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

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.

arXiv commentREVTeX4, 16 pages, 4 figures. v2: some additions and corrections

References in corpus (18)

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Spectral Density of the QCD Dirac Operator near Zero Virtuality
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• Random Matrix Theory of a Chaotic Andreev Quantum Dot
• The spectrum of the Dirac operator near zero virtuality for $N_c=2$
• Finite volume partition functions and Itzykson-Zuber integrals
• Vicious walkers, friendly walkers and Young tableaux II: With a wall
• Kinetics of a Diffusive Capture Process: Lamb Besieged by a Pride of Lions
• Scaling limit of vicious walks and two-matrix model
• Exact Asymptotics for One-Dimensional Diffusion with Mobile Traps
• Universality in Chiral Random Matrix Theory at $\beta =1$ and $\beta =4$
• Dynamical Correlations among Vicious Random Walkers
• Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries
• Families of Vicious Walkers
• Ordering of Random Walks: The Leader and the Laggard
• Dynamical Correlations for Vicious Random Walk with a Wall
• Directed Percolation and Generalized Friendly Walkers
• Moments of vicious walkers and Möbius graph expansions

Cited by in corpus (23)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Non-intersecting Brownian Interfaces and Wishart Random Matrices
• Random matrix theory and symmetric spaces
• A note on biorthogonal ensembles
• Nonintersecting Brownian excursions
• Noncolliding Brownian Motion and Determinantal Processes
• Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
• Functional central limit theorems for vicious walkers
• Orbit measures, random matrix theory and interlaced determinantal processes
• Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• Correlation Functions of XX0 Heisenberg Chain, q-Binomial Determinants, and Random Walks
• Watermelon configurations with wall interaction: exact and asymptotic results
• The correlation functions of the XX Heisenberg magnet and random walks of vicious walkers
• The Ising Limit of the XXZ Heisenberg Magnet and Certain Thermal Correlation Functions
• Determinantal process starting from an orthogonal symmetry is a Pfaffian process
• Scaling of many-particle correlations in a dissipative sandpile
• Preserving Topology while Breaking Chirality: From Chiral Orthogonal to Anti-symmetric Hermitian Ensemble
• Universal Broadening of Zero Modes: A General Framework and Identification
• Bessel process, Schramm-Loewner evolution, and Dyson model
• Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes
• Noncolliding Brownian motions and Harish-Chandra formula
• Extreme value distributions of noncolliding diffusion processes