The correlation functions of the XX Heisenberg magnet and random walks of vicious walkers

Abstract

A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the XX Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called "vicious" random walkers (the vicious walkers annihilate each other provided they arrive at the same lattice site). It is shown that the two-point correlation function of spins, calculated over eigen-states of the XX magnet, can be interpreted as the generating function of paths made by a single walker in a medium characterized by a non-constant number of vicious neighbors. The answers are obtained for a number of paths made by the described walker from some fixed lattice site to another sufficiently remote one. Asymptotical estimates for the number of paths are provided in the limit, when the number of steps is increased.

16 pages, 1 figure, LaTeX. Extended talk at the Conference "Classical and Quantum Integrable Systems CQIS-08" (Protvino, Russia, January 21-24, 2008). To appear in Theoretical and Mathematical Physics in 2009 (In Russian)

References in corpus (4)

• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory
• Functional Integration with "Automorphic" Boundary Conditions and Correlators of Z-Components of Spins in the XY and XX Heisenberg Chains
• Random turn walk on a half line with creation of particles at the origin

Cited by in corpus (8)

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• Correlation Functions as Nests of Self-Avoiding Paths
• Scaling of many-particle correlations in a dissipative sandpile
• Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions
• Combinatorial Interpretation of the Scalar Products of State Vectors of Integrable Models
• Spin correlation functions, Ramus-like identities, and enumeration of constrained lattice walks and plane partitions
• The ground state-vector of the XY Heisenberg chain and the Gauss decomposition