Geodesic
Some explicit results for the generalized emptiness formation probability of the six-vertex model
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 24, Tome 465 (2017), pp. 157-173 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We study a multi-point correlation function of the six-vertex model on the square lattice with the domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes probability of observing the ferroelectric order around all the vertices of any Ferrer diagram $\lambda$ at the top-left corner of the lattice. For the free-fermion model we derive and compare explicit formulas for this correlation function for two cases of diagram $\lambda$: the square and triangle. We found a connection of our formulas with the $\tau$-function of the sixth Painlevé equation. 
@article{ZNSL_2017_465_a9,
     author = {A. V. Kitaev and A. G. Pronko},
     title = {Some explicit results for the generalized emptiness formation probability of the six-vertex model},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {157--173},
     year = {2017},
     volume = {465},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_465_a9/}
}
TY  - JOUR
AU  - A. V. Kitaev
AU  - A. G. Pronko
TI  - Some explicit results for the generalized emptiness formation probability of the six-vertex model
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 157
EP  - 173
VL  - 465
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_465_a9/
LA  - en
ID  - ZNSL_2017_465_a9
ER  - 
%0 Journal Article
%A A. V. Kitaev
%A A. G. Pronko
%T Some explicit results for the generalized emptiness formation probability of the six-vertex model
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 157-173
%V 465
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_465_a9/
%G en
%F ZNSL_2017_465_a9
A. V. Kitaev; A. G. Pronko. Some explicit results for the generalized emptiness formation probability of the six-vertex model. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 24, Tome 465 (2017), pp. 157-173. http://geodesic.mathdoc.fr/item/ZNSL_2017_465_a9/

[1] V. E. Korepin, “Calculations of norms of Bethe wave functions”, Commun. Math. Phys., 86 (1982), 391–418 | DOI | MR | Zbl

[2] A. G. Izergin, “Partition function of the six-vertex model in the finite volume”, Sov. Phys. Dokl., 32 (1987), 878–879 | MR | Zbl

[3] A. G. Izergin, D. A. Coker, V. E. Korepin, “Determinant formula for the six-vertex model”, J. Phys. A, 25 (1992), 4315–4334 | DOI | MR | Zbl

[4] K. Eloranta, “Diamond ice”, J. Stat. Phys., 96 (1999), 1091–1109 | DOI | MR | Zbl

[5] P. Zinn-Justin, The influence of boundary conditions in the six-vertex model, arXiv: cond-mat/0205192

[6] F. Colomo, A. Sportiello, “Arctic curves of the six-vertex model on generic domains: the Tangent Method”, J. Stat. Phys., 164 (2016), 1488–1523 | DOI | MR | Zbl

[7] V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[8] N. M. Bogoliubov, C. L. Malyshev, “Integrable models and combinatorics”, Russian Math. Surveys, 70:5 (2015), 789–856 | DOI | MR | Zbl

[9] N. M. Bogoliubov, A. V. Kitaev, M. B. Zvonarev, “Boundary polarization in the six-vertex model”, Phys. Rev. E, 65 (2002), 026126 | DOI | MR

[10] N. M. Bogoliubov, A. G. Pronko, M. B. Zvonarev, “Boundary correlation functions of the six-vertex model”, J. Phys. A, 35 (2002), 5525–5541 | DOI | MR | Zbl

[11] F. Colomo, A. G. Pronko, “Emptiness formation probability in the domain-wall six-vertex model”, Nucl. Phys. B, 798 (2008), 340–362 | DOI | MR | Zbl

[12] F. Colomo, A. G. Pronko, “The arctic curve of the domain-wall six-vertex model”, J. Stat. Phys., 138 (2010), 662–700 | DOI | MR | Zbl

[13] A. G. Pronko, “On the emptiness formation probability in the free-fermion six-vertex model with domain wall boundary conditions”, J. Math. Sci. (N.Y.), 192:1 (2013), 101–116 | DOI | MR | Zbl

[14] F. Colomo, A. G. Pronko, “Third-order phase transition in random tilings”, Phys. Rev. E, 88 (2013), 042125 | DOI

[15] F. Colomo, A. G. Pronko, “Thermodynamics of the six-vertex model in an $L$-shaped domain”, Comm. Math. Phys., 339 (2015), 699–728 | DOI | MR | Zbl

[16] A. V. Kitaev, A. G. Pronko, “Emptiness formation probability of the six-vertex model and the sixth Painlevé equation”, Comm. Math. Phys., 345 (2016), 305–354. | DOI | MR | Zbl

[17] F. Colomo, A. G. Pronko, A. Sportiello, “Generalized emptiness formation probability in the six-vertex model”, J. Phys. A, 49 (2016), 415203 | DOI | MR | Zbl

[18] P. L. Ferrari, B. Vető, “The hard-edge tacnode process for Brownian motion”, Electron. J. Probab., 22 (2017), 79, 32 pp. | DOI | MR | Zbl

[19] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, San Diego, CA, 1982 | MR | Zbl

[20] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn., Oxford University Press, Oxford, 1995 | MR | Zbl

[21] A. Erdelyi, Higher transcendental functions, v. 1, McGraw-Hill, New York, 1953 | Zbl

[22] M. Jimbo, T. Miwa, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Physica D, 2 (1981), 407–448 | DOI | MR | Zbl

[23] K. Okamoto, “Studies on the Painlevé equations. I. Sixth Painleve Equation $\mathrm P_\mathrm{VI}$”, Ann. Mat. Pura Appl., 146 (1987), 337–381 | DOI | MR | Zbl

[24] P. J. Forrester, N. S. Witte, “Application of the $\tau$-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits”, Nagoya Math. J., 174 (2004), 29–114 | DOI | MR | Zbl

[25] N. J. A. Sloane, Sequence A106729, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A106729