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.
Keywords:
random walk, Heisenberg magnet, correlation function.
Citation:
N. M. Bogolyubov, K. L. Malyshev, “Correlation functions of the XX Heisenberg magnet and random walks of vicious walkers”, TMF, 159:2 (2009), 179–193; Theoret. and Math. Phys., 159:2 (2009), 563–574
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\pages 179--193
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\jour Theoret. and Math. Phys.
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\vol 159
\issue 2
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Linking options:
https://www.mathnet.ru/eng/tmf6341
https://doi.org/10.4213/tmf6341
https://www.mathnet.ru/eng/tmf/v159/i2/p179
This publication is cited in the following 10 articles:
N. M. Bogoliubov, C. L. Malyshev, “Semi-infinite Heisenberg XX0 chain and random walks”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 30, Zap. nauchn. sem. POMI, 532, POMI, SPb., 2024, 91–108
C Malyshev, N M Bogoliubov, “Spin correlation functions, Ramus-like identities, and enumeration of constrained lattice walks and plane partitions”, J. Phys. A: Math. Theor., 55:22 (2022), 225002
J. Math. Sci. (N. Y.), 242:5 (2019), 628–635
J. Math. Sci. (N. Y.), 238:6 (2019), 779–792
N. M. Bogolyubov, K. L. Malyshev, “Integrable models and combinatorics”, Russian Math. Surveys, 70:5 (2015), 789–856
J. Math. Sci. (N. Y.), 216:1 (2016), 8–22
J. Math. Sci. (N. Y.), 200:6 (2014), 662–670
Bogoliubov N.M., Malyshev C., “Correlation Functions of Xxo Heisenberg Chain, Q-Binomial Determinants, and Random Walks”, Nucl. Phys. B, 879 (2014), 268–291
N. M. Bogolyubov, K. L. Malyshev, “Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions”, Theoret. and Math. Phys., 169:2 (2011), 1517–1529
N. M. Bogoliubov, K. Malyshev, “The correlation functions of the XXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers”, St. Petersburg Math. J., 22:3 (2011), 359–377