Lieb's Square Ice Constant

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.

REFERENCES:

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."