Geodesic
A combinatorial interpretation of the scalar products of state vectors of integrable models
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 33-46 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The representation of Bethe wave functions of certain integrable models via Schur functions allows one to apply the well-developed theory of symmetric functions to the calculation of thermal correlation functions. The algebraic relations arising in the calculation of scalar products and correlation functions are based on the Binet–Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of Bethe state vectors in terms of nests of self-avoiding lattice paths constituting so-called watermelon configurations. The proposed interpretation is, in turn, related to the enumeration of boxed plane partitions. 
@article{ZNSL_2014_421_a2,
     author = {N. M. Bogoliubov and C. Malyshev},
     title = {A combinatorial interpretation of the scalar products of state vectors of integrable models},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {33--46},
     year = {2014},
     volume = {421},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a2/}
}
TY  - JOUR
AU  - N. M. Bogoliubov
AU  - C. Malyshev
TI  - A combinatorial interpretation of the scalar products of state vectors of integrable models
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2014
SP  - 33
EP  - 46
VL  - 421
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a2/
LA  - en
ID  - ZNSL_2014_421_a2
ER  - 
%0 Journal Article
%A N. M. Bogoliubov
%A C. Malyshev
%T A combinatorial interpretation of the scalar products of state vectors of integrable models
%J Zapiski Nauchnykh Seminarov POMI
%D 2014
%P 33-46
%V 421
%U http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a2/
%G en
%F ZNSL_2014_421_a2
N. M. Bogoliubov; C. Malyshev. A combinatorial interpretation of the scalar products of state vectors of integrable models. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 33-46. http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a2/

[1] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford, 1995 | MR | Zbl

[2] W. Fulton, Young Tableaux with Application to Representation Theory and Geometry, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[3] D. M. Bressoud, Proofs and Confirmations. The Story of the Alternating Sign Matrix Conjecture, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[4] G. Schehr, S. N. Majumdar, A. Comtet, P. J. Forrester, “Reunion probability of $N$ vicious walkers: typical and large fluctuations for large $N$”, J. Stat. Phys., 149 (2012), 385–410 | DOI | MR | Zbl

[5] P. Zinn-Justin, “Six-vertex model with domain wall boundary conditions and one-matrix model”, Phys. Rev. E, 62 (2000), 3411–3418 | DOI | MR

[6] A. Okounkov, “Symmetric functions and random partitions”, Symmetric Functions 2001, Surveys of Developments and Perspectives, NATO Science Series, 74, 2002, 223–252 | MR | Zbl

[7] K. Hikami, T. Imamura, “Vicious walkers and hook Young tableaux”, J. Phys. A: Math. Gen., 36 (2003), 3033–3048 | DOI | MR | Zbl

[8] A. Okounkov, N. Reshetikhin, “Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram”, J. Amer. Math. Soc., 16 (2003), 581–603 | DOI | MR | Zbl

[9] G. Téllez, P. J. Forrester, “Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma”, J. Stat. Phys., 148 (2012), 824–855 | DOI | MR | Zbl

[10] N. M. Bogoliubov, “Boxed plane partitions as an exactly solvable boson model”, J. Phys. A: Math. Gen., 38 (2005), 9415–9430 | DOI | MR | Zbl

[11] N. M. Bogoliubov, J. Timonen, “Correlation functions for a strongly coupled boson system and plane partitions”, Phil. Trans. Roy. Soc. A, 369 (2011), 1319–1333 | DOI | MR | Zbl

[12] N. M. Bogoliubov, “XX0 Heisenberg chain and random walks”, J. Math. Sci., 138 (2006), 5636–5643 | DOI | MR

[13] N. M. Bogoliubov, “The integrable models for the vicious and friendly walkers”, J. Math. Sci., 143 (2007), 2729–2737 | DOI | MR

[14] N. M. Bogoliubov, C. Malyshev, “The correlation functions of the XX Heisenberg magnet and random walks of vicious walkers”, Theor. Math. Phys., 159 (2009), 563–574 | DOI | MR | Zbl

[15] N. M. Bogoliubov, C. Malyshev, “The correlation functions of the XXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicioius walkers”, St. Petersburg Math. J., 22:3 (2011), 359–377 | DOI | MR | Zbl

[16] N. M. Bogoliubov, C. Malyshev, Correlation functions of the XXZ chain at zero anisotropy and enumeration of boxed plane partitions, PDMI preprint 19/2012 http://www.pdmi.ras.ru/preprint/2012/12-19.html

[17] L. D. Faddeev, “Quantum completely integrable models of field theory”, Sov. Sci. Rev. Math. C, 1 (1980), 107–160; 40 Years in Mathematical Physics, v. 2, World Sci. Ser. 20th Century Math., World Sci., Singapore, 1995, 187–235 | DOI | MR

[18] V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[19] G. Kuperberg, “Another proof of the alternating sign martrix conjecture”, Int. Math. Res. Notices, 1996 (1996), 139–150 | DOI | MR | Zbl

[20] A. Klimyk, K. Schmudgen, Quantum Groups and their Representations, Springer, Berlin, 1997 | MR | Zbl

[21] I. Gessel, X. G. Viennot, Determinants, paths, and plane partitions, Preprint, 1989

[22] I. Gessel, G. Viennot, “Binomial determinants, paths, and hook length formulae”, Adv. Math., 58 (1985), 300–321 | DOI | MR | Zbl

[23] A. J. Guttmann, A. L. Owczarek, X. G. Viennot, “Vicious walkers and Young tableaux I: without walls”, J. Phys. A: Math. Gen., 31 (1998), 8123–8135 | DOI | MR | Zbl