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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 5(425), Pages 3–74 (Mi umn9651)  

This article is cited in 10 scientific papers (total in 10 papers)

Integrable models and combinatorics

N. M. Bogolyubovab, K. L. Malysheva

a St. Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences
b St. Petersburg National Research University of Information Technology, Mechanics, and Optics

Abstract: Relations between quantum integrable models solvable by the quantum inverse scattering method and some aspects of enumerative combinatorics and partition theory are discussed. The main example is the Heisenberg $XXZ$ spin chain in the limit cases of zero or infinite anisotropy. Form factors and some thermal correlation functions are calculated, and it is shown that the resulting form factors in a special $q$-parametrization are the generating functions for plane partitions and self-avoiding lattice paths. The asymptotic behaviour of the correlation functions is studied in the case of a large number of sites and a moderately large number of spin excitations. For sufficiently low temperature a relation is established between the correlation functions and the theory of matrix integrals.
Bibliography: 125 titles.

Keywords: correlation functions, Heisenberg magnet, four-vertex model, plane partitions, generating functions, symmetric functions.

Funding Agency Grant Number
Russian Science Foundation 14-11-00598
This work was supported by the Russian Science Foundation (grant no. 14-11-00598).


DOI: https://doi.org/10.4213/rm9651

Full text: PDF file (1492 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2015, 70:5, 789–856

Bibliographic databases:

UDC: 517.958+530.145
PACS: 02.10.Os; 03.65.-w
MSC: Primary 82B20, 37K60, 05E05; Secondary 82B30, 82B41, 82D40, 05C81
Received: 31.01.2015

Citation: N. M. Bogolyubov, K. L. Malyshev, “Integrable models and combinatorics”, Uspekhi Mat. Nauk, 70:5(425) (2015), 3–74; Russian Math. Surveys, 70:5 (2015), 789–856

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. J. Math. Sci. (N. Y.), 224:2 (2017), 199–213  mathnet  crossref  mathscinet
    2. Nicolay M. Bogoliubov, Cyril Malyshev, “Zero Range Process and Multi-Dimensional Random Walks”, SIGMA, 13 (2017), 056, 14 pp.  mathnet  crossref
    3. N. Bogoliubov, “Continuous time multidimensional walks as an integrable model”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 24, Zap. nauchn. sem. POMI, 465, POMI, SPb., 2017, 13–26  mathnet
    4. N. Bogoliubov, C. Malyshev, “Correlation functions as nests of self-avoiding paths”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 24, Zap. nauchn. sem. POMI, 465, POMI, SPb., 2017, 27–45  mathnet
    5. A. V. Kitaev, A. G. Pronko, “Some explicit results for the generalized emptiness formation probability of the six-vertex model”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 24, Zap. nauchn. sem. POMI, 465, POMI, SPb., 2017, 157–173  mathnet
    6. M. Saeedian, A. Zahabi, “Phase structure of XX0 spin chain and nonintersecting Brownian motion”, J. Stat. Mech.-Theory Exp., 2018, 013104, 36 pp.  crossref  mathscinet  isi  scopus
    7. N. Bogoliubov, C. Malyshev, “The phase model and the norm-trace generating function of plane partitions”, J. Stat. Mech.-Theory Exp., 2018, 083101  crossref  isi  scopus
    8. C. L. Malyshev, N. M. Bogolyubov, “The ground state-vector of the $XY$ Heisenberg chain and the Gauss decomposition”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 25, K 70-letiyu M. A. Semenova-Tyan-Shanskogo, Zap. nauchn. sem. POMI, 473, POMI, SPb., 2018, 66–76  mathnet
    9. C. L. Malyshev, N. M. Bogolyubov, “The partition function of the four-vertex model in a special external field”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 25, K 70-letiyu M. A. Semenova-Tyan-Shanskogo, Zap. nauchn. sem. POMI, 473, POMI, SPb., 2018, 77–84  mathnet
    10. G. P. Pron'ko, A. G. Pronko, “Off-shell Bethe states and the six-vertex model”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 25, K 70-letiyu M. A. Semenova-Tyan-Shanskogo, Zap. nauchn. sem. POMI, 473, POMI, SPb., 2018, 228–243  mathnet
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