TOPICS
Search

Twenty-Vertex Entropy Constant


Let L denote the n×n triangular lattice with wraparound. Call an orientation of L an assignment of a direction to each edge of L, and denote the number of orientations of L such that each vertex has three inwardly directed and three outwardly directly edges by g_n. Such an orientation is said to consist of Eulerian orientation.

Baxter showed that

lim_(n->infty)g_n^(1/n^2)=sqrt((27)/4)
(1)
=3/2sqrt(3)
(2)
=2.5980762113...
(3)

(OEIS A104956; Finch 2003, p. 412), which is known as the twenty-vertex entropy constant (a term coined here for the first time).


See also

Lieb's Square Ice Constant

Explore with Wolfram|Alpha

References

Baxter, R. J. "F Model on a Triangular Lattice." J. Math. Phys. 10, 1211-1216, 1969.Baxter, R. J. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.Finch, S. R. "Lieb's Square Ice Constant." §5.24 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 412-413, 2003.Sloane, N. J. A. Sequence A104956 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Twenty-Vertex Entropy Constant

Cite this as:

Weisstein, Eric W. "Twenty-Vertex Entropy Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Twenty-VertexEntropyConstant.html

Subject classifications