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Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov-de Gennes random matrices

Makoto Katori1, Hideki Tanemura, Taro Nagao

  • 1Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan. katori@phys.chou-u.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2003
PubMed

Insights

This study analyzes one-dimensional vicious walkers, revealing their connection to Brownian meanders and random matrix theory. The findings link particle-hole symmetric Hamiltonians to walker distributions and symmetry class transitions.

Area of Science:

  • Statistical Mechanics
  • Stochastic Processes
  • Condensed Matter Physics

Background:

  • Investigates spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restrictions.
  • Establishes equivalence between the continuum version of vicious walkers and noncolliding Brownian meanders.

Purpose of the Study:

  • To explore the mathematical connections between vicious walkers, Brownian meanders, and random matrix theory.
  • To characterize the spatial distribution and time evolution of these systems.

Main Methods:

  • Utilizes the concept of Brownian meanders, a transform of Bessel processes, to model walker behavior.
  • Applies eigenvalue statistics of Gaussian ensembles from Bogoliubov-de Gennes Hamiltonians (superconductivity mean-field theory) to describe spatial distributions.
  • Analyzes symmetry class transitions (Type C to CI) in random matrix theory for time evolution.

Main Results:

  • Demonstrates that the spatial distribution of wall-restricted vicious walkers corresponds to eigenvalue statistics of particle-hole symmetric Gaussian ensembles.
  • Shows that the time evolution is characterized by a symmetry class change from Type C to Type CI.
  • Clarifies the relationship between generalized meanders (even-dimensional Bessel processes) and chiral random matrix theory.

Conclusions:

  • The study provides a comprehensive framework linking vicious walkers to advanced concepts in stochastic processes and random matrix theory.
  • Highlights the role of symmetry classes in describing the dynamics of these complex systems.
  • Offers insights into the statistical properties of noncolliding particle systems relevant to superconductivity and quantum chaos.

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