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Correlation functions of the XX Heisenberg magnet and random walks of vicious walkers

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Abstract

We investigate a relation between random walks on a one-dimensional periodic lattice and correlation functions of the XX Heisenberg spin chain. Operator averages over the ferromagnetic state play the role of generating functions of the number of paths traveled by so-called vicious random walkers (vicious walkers annihilate each other if they arrive at the same lattice site). We show that the two-point correlation function of spins calculated over eigenstates of the XX magnet can be interpreted as the generating function of paths traveled by a single walker in a medium characterized by a variable number of vicious neighbors. We obtain answers for the number of paths traveled by the described walker from a fixed lattice site to a sufficiently remote site. We provide asymptotic estimates of the number of paths in the limit of a large number of steps.

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Authors and Affiliations

  1. St. Petersburg Department of Steklov Institute of Mathematics, RAS, St. Petersburg, Russia

    N. M. Bogoliubov & C. Malyshev

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  1. N. M. Bogoliubov
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  2. C. Malyshev
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Correspondence to N. M. Bogoliubov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 2, pp. 179–193, May, 2009.

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Bogoliubov, N.M., Malyshev, C. Correlation functions of the XX Heisenberg magnet and random walks of vicious walkers. Theor Math Phys 159, 563–574 (2009). https://doi.org/10.1007/s11232-009-0046-y

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