The five-vertex model and enumerations of plane partitions

A. G. Pronko

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 23, Tome 433 (2015), pp. 204-223

Résumé

We consider the five-vertex model on an $N\times2N$ lattice with fixed boundary conditions of a special type. We discuss a determinantal formula for the partition function in application to description of various enumrations of $N\times N\times (M-N)$ boxed plane partions. It is shown, that at the free-fermion point of the model, this formula reproduces MacMahon formula for the number of boxed plane partitions, while for generic weights (out of the free-fermion point) it describes enumerations with the weight depending on the cumulative number of jumps along vertical (or horisontal) rows. Various representations for the partition function, which describes such enumerations, are obtained.

Bibliographie

[1] L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi i XYZ model Geizenberga”, UMN, 34:5 (1979), 13–63 | MR

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

[3] P. P. Kulish, “Quantum difference nonlinear Shroedinger equation”, Lett. Math. Phys., 5 (1981), 191–197 | DOI | MR | Zbl

[4] V. S. Gerdjikov, M. I. Ivanov, P. P. Kulish, “Expansions over the “squared” solutions and difference evolution equations”, J. Math. Phys., 25 (1984), 25–34 | DOI | MR

[5] N. M. Bogoliubov, R. K. Bullough, G. D. Pang, “Exact solution of a $q$-boson hopping model”, Phys. Rev. B, 47 (1993), 11945–11498 | DOI

[6] N. M. Bogolyubov, P. P. Kulish, “Tochno-reshaemye modeli kvantovoi nelineinoi optiki”, Zap. nauchn. semin. POMI, 398, 2012, 26–54 | MR

[7] N. M. Bogoliubov, T. Nasar, “On the spectrum of the non-Hermitian phase-difference model”, Phys. Lett. A, 234 (1997), 345–350 | DOI | MR | Zbl

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

[9] A. G. Izergin, “Statsumma shestivershinnoi modeli v konechnom ob'eme”, DAN, 297 (1987), 331–334 | MR

[10] 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

[11] G. Kuberberg, “Another proof of the alternating-sign matrix conjecture”, Int. Res. Math. Notices, 1996 (1996), 139–150 | DOI | MR

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

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

[14] N. M. Bogolyubov, “Chetyrekhvershinnaya model i sluchainye ukladki”, Teor. mat. fiz., 155:1 (2008), 25–38 | DOI | MR | Zbl

[15] N. M. Bogoliubov, C. Malyshev, “Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks”, Nucl. Phys. B, 879 (2014), 268–291 | DOI | MR | Zbl

[16] N. M. Bogolyubov, “Skalyarnye proizvedeniya vektorov sostoyanii v polnostyu asimmetrichnykh tochno reshaemykh modelyakh na koltse”, Zap. nauchn. semin. POMI, 398, 2012, 5–25 | MR

[17] 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

[18] F. Colomo, A. G. Pronko, “The limit shape of large alternating-sign matrices”, SIAM J. Discrete Math., 24 (2010), 1558–1571 | DOI | MR | Zbl

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

[20] F. Colomo, A. G. Pronko, “Thermodynamics of the six-vertex model in an $L$-shaped domain”, Comm. Math. Phys., 339:2 (2015), 699–728 | MR | Zbl

[21] N. M. Bogolyubov, “Pyativershinnaya model s fiksirovannymi granichnymi usloviyami”, Algebra i analiz, 21:3 (2009), 58–78 | MR | Zbl

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

[23] G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing, 1976 | MR | Zbl

[24] V. S. Kapitonov, A. G. Pronko, “Pyativershinnaya model i ploskie razbieniya v yaschike”, Zap. nauchn. semin. POMI, 360, 2008, 162–179 | MR | Zbl

[25] V. S. Kapitonov, A. G. Pronko, “Vzveshennye perechisleniya ploskikh razbienii v yaschike i neodnorodnaya pyativershinnaya model”, Zap. nauchn. semin. POMI, 398, 2012, 125–144 | MR

[26] H. Cohn, M. Larsen, J. Propp, “The shape of a typical boxed plane partition”, New York J. Math., 4 (1998), 137–165 | MR | Zbl

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

[28] A. G. Pronko, “O veroyatnosti obrazovaniya pustoty v svobodnofermionnoi shestivershinnoi modeli s granichnymi usloviyami domennoi stenki”, Zap. nauchn. semin. POMI, 398, 2012, 179–208 | MR

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

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

[31] 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 | MR | Zbl