Let
denote the
square lattice with wraparound. Call an orientation of an assignment of a direction to each edge of , and denote the number of orientations of such that each vertex has two inwardly directed and two outwardly
directly edges by . Such an orientation is said to obey the ice rule, or to
consist of Eulerian orientation. For , 2, ..., the first few values of are 4, 18, 148, 2970, ... (OEIS A054759).
Lieb showed that
(1)
(2)
(3)
(OEIS A118273; Finch 2003, p. 412), which is known as Lieb's square ice constant, also known as the square ice constant, residual
entropy for square ice, and six-vertex entropy model.
Baxter, R. J. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.Bell,
G. M. and Lavis, D. A. Statistical Methods of Lattice Systems, Vol. 1.
New York: Springer-Verlag, 1999.Bell, G. M. and Lavis, D. A.
Statistical
Methods of Lattice Systems, Vol. 2. New York: Springer-Verlag, 1999.Finch,
S. R. "Lieb's Square Ice Constant." §5.24 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 412-413,
2003.Godsil, C.; Grötschel, M.; and Welch, D. J. A. "Combinatorics
in Statistical Physics." In Handbook
of Combinatorics, Vol. 2 (Ed. R. L. Graham, M. Grötschel,
and L. Lovász). Cambridge, MA: MIT Press, pp. 1925-1954, 1995.Lieb,
E. H. "The Residual Entropy of Square Ice." Phys. Rev.162,
162-172, 1967.Lieb, E. H. "Exact Solution of the Problem of
the Entropy of Two-Dimensional Ice." Phys. Rev. Lett.18, 692-694,
1967.Lieb, E. H. and Wu, F. Y. "Two-Dimensional Ferroelectric
Models." In Phase Transitions and Critical Phenomena, Vol. 1 (Ed.
C. Domb and M. S. Greene). New York: Academic Press, pp. 331-490,
1972.Percus, J. K. Combinatorial
Methods. New York: Springer-Verlag, 1971.Sloane, N. J. A.
Sequences A054759 and A118273
in "The On-Line Encyclopedia of Integer Sequences."
denote the 
square lattice with wraparound. Call an orientation of 
an assignment of a direction to each edge of 
, and denote the number of orientations of 
such that each vertex has two inwardly directed and two outwardly
directly edges by 
. Such an orientation is said to obey the ice rule, or to
consist of Eulerian orientation. For 
, 2, ..., the first few values of 
are 4, 18, 148, 2970, ... (OEIS A054759).
