Abstract
We consider the five-vertex model on a square lattice with fixed boundary conditions which corresponds to weighted (with weight q per elementary cube) enumerations of boxed plane partitions. We calculate the one-point correlation function of the model which describes the probability of a given state on an edge (polarization). This generalizes an analogous result obtained previously by the authors for unweighted (weighted with weight q = 1) enumerations of plane partitions. Bibliography: 14 titles.
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References
G. E. Andrews, The Theory of Partitions, Addison-Wesley Publ. (1976).
D. M. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge Univ. Press, Cambridge (1999).
H. Cohn, M. Larsen, and J. Propp, “The shape of a typical boxed plane partition,” New York J. Math., 4, 137–165 (1998).
A. Borodin, V. Gorin, and E. M. Rains, “q-Distributions on boxed plane partitions,” Selecta Math. (N. S.), 16, 731–789 (2010).
N. M. Bogoliubov, “Boxed plane partitions as an exactly solvable boson model,” J. Phys. A, 38, 9415–9430 (2005).
N. M. Bogoliubov, “Four-vertex model and random tilings,” Teor. Mat. Fiz., 155, 25–38 (2008).
V. S. Kapitonov and A. G. Pronko, “The five-vertex model and boxed plane partitions,” J. Math. Sci., 158, 858–867 (2009).
F. A. Berezin, The Method of Second Quantization [in Russian], 2nd ed. (aug.), Moscow (1986).
R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q -Analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin (2010).
D. S. Moak, “The q-analogue of the Laguerre polynomials,” J. Math. Anal. Appl., 81, 20–47 (1981).
R. Koekoek, “Generalizations of a q-analogue of Laguerre polynomials,” J. Approx. Theory, 69, 55–83 (1992).
S. G. Moreno and E. M. García-Caballero, “q-Sobolev orthogonality of the q-Laguerre polynomials \(\{L_n^{(-N)}(\cdot;q)\}_{n=0}^\infty\) for positive integers N,” J. Korean Math. Soc., 48, 913–926 (2011).
G. Szegö, Orthogonal Polynomials, American Colloquium Publications, Vol. XXIII, 4th ed., Amer. Math. Soc., Providence, RI (1975).
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Univ. Press, Oxford (1995).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 398, 2012, pp. 125–144.
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Kapitonov, V.S., Pronko, A.G. Weighted enumerations of boxed plane partitions and the inhomogeneous five-vertex model. J Math Sci 192, 70–80 (2013). https://doi.org/10.1007/s10958-013-1374-x
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DOI: https://doi.org/10.1007/s10958-013-1374-x