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Correlation Functions as Nests of Self-Avoiding Paths

  • N. BogoliubovEmail author
  • C. Malyshev
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We discuss a connection between the XXZ Heisenberg spin chain in the limiting case of zero anisotropy and some aspects of enumerative combinatorics. The representation of the Bethe wave functions in terms of Schur functions allows us to apply the theory of symmetric functions to calculating correlation functions. We provide a combinatorial derivation of the dynamical correlation functions of the projection operator in terms of nests of self-avoiding lattice paths.

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References

  1. 1.
    R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge (1999).Google Scholar
  2. 2.
    C. Williams, Explorations in Quantum Computing, Springer (2010).Google Scholar
  3. 3.
    E. Fama, “Random walks in stock market prices,” Fin. Anal. J., 51, 75–80 (1995).CrossRefGoogle Scholar
  4. 4.
    K. Sneppen, Models of Life, Cambridge Univ. Press (2014).Google Scholar
  5. 5.
    G. Fischer and D. Laming, Contributions to Mathematical Psychology, Psychometrics, and Methodology, Springer (2012).Google Scholar
  6. 6.
    S. Redner, A Guide to First-Passage Processes, Cambridge Univ. Press (2001).Google Scholar
  7. 7.
    E. Renshaw, Stochastic Population Processes, Oxford Univ. Press (2011).Google Scholar
  8. 8.
    M. Fisher, “Walks, walls, wetting and melting,” J. Stat. Phys., 34, 667–730 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Forrester, “Exact solution of the lock step model of vicious walkers,” J. Phys. A, 23, 1259 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    T. Nagao and P. Forrester, “Vicious random walkers and a discretization of Gaussian random matrix ensembles,” Nucl. Phys. B, 620, 551–565 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Guttmann, A. Owczarek, and X. Viennot, “Vicious walkers and Young tableaux I: without walls,” J. Phys. A, 31, 8123 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    C. Krattenthaler, A. Guttmann, and X. Viennot, “Vicious walkers, friendly walkers and young tableaux: II. With a wall,” J. Phys. A, 33, 8835 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Katori and H. Tanemura, “Scaling limit of vicious walks and two-matrix model,” Phys. Rev. E, 66, 011105 (2002).Google Scholar
  14. 14.
    N. Bogoliubov, “XX Heisenberg chain and random walks,” J. Math. Sci., 138, 5636–5643 (2006).MathSciNetCrossRefGoogle Scholar
  15. 15.
    I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press (1995).Google Scholar
  16. 16.
    L. Faddeev, “Quantum completely integrable models in field theory,” Sov. Rev. Sci. C: Math. Phys., 1, 107–155 (1980).MathSciNetzbMATHGoogle Scholar
  17. 17.
    V. Korepin, N. Bogoliubov, and A. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge (1993).CrossRefzbMATHGoogle Scholar
  18. 18.
    N. Bogoliubov and C. Malyshev, “Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks,” Nucl. Phys. B, 879, 268–291 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    N. Bogoliubov and C. Malyshev, “Integrable models and combinatorics,” Russian Math. Surveys, 70, 789–856 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    N. Bogoliubov and C. Malyshev, “Multi-dimensional random walks and integrable phase models,” J. Math. Sci., 224, 199–213 (2017).CrossRefzbMATHGoogle Scholar
  21. 21.
    N. Bogoliubov and C. Malyshev, “Zero range process and multi-dimensional random walks,” SIGMA, 13, Paper 056 (2017).Google Scholar
  22. 22.
    B.-Q. Jin and V. E. Korepin, “Entanglement, Toeplitz determinants and Fisher–Hartwig conjecture,” J. Stat. Phys., 116, 79–95 (2004).CrossRefzbMATHGoogle Scholar
  23. 23.
    N. Bogoliubov and C. Malyshev, “Correlation functions of the XX Heisenberg magnet and random walks of vicious walkers,” Theor. Math. Phys., 159, 563–574 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    W. Fulton, Young Tableaux. With Applications to Representation Theory and Geometry, Cambridge Univ. Press (1997).Google Scholar
  25. 25.
    D. J. Gross and E. Witten, “Possible third-order phase transition in the large-N lattice gauge theory,” Phys. Rev. D, 21, 446 (1980).CrossRefGoogle Scholar
  26. 26.
    K. Johansson, “Unitary random matrix model,” Math. Res. Lett., 5, 63–82 (1998).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St.Petersburg Department of Steklov Institute of MathematicsITMO UniversitySt.PetersburgRussia

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