Lieb's square ice constant is the exponential growth constant for ice-rule orientations of the square point
lattice. Let denote the square point lattice
with wraparound. For , retain the two loops, and for , retain the parallel edges produced by 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
directed edges by . Such an orientation is said to obey the ice rule, or to
be an Eulerian orientation. For , 2, ..., the first few values of are 4, 18, 148, 2970, ... (OEIS A054759).
Lieb (1967) showed that
(1)
(2)
(3)
(OEIS A118273; Finch 2003, p. 412), also called the square ice constant. Its natural logarithm
is the residual entropy per vertex, in units of the Boltzmann constant, for the square
ice model (Lieb 1967).
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 point lattice
with wraparound. For 
, retain the two loops, and for 
, retain the parallel edges produced by 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
directed edges by 
. Such an orientation is said to obey the ice rule, or to
be an Eulerian orientation. For 
, 2, ..., the first few values of 
are 4, 18, 148, 2970, ... (OEIS A054759).
