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Twenty-Vertex Entropy Constant

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The term twenty-vertex entropy constant is used here for the asymptotic growth constant of Eulerian orientations of the triangular lattice. Let L be the lattice graph on an n×n period of the triangular lattice with opposite sides identified, where n>=3. Let g_n count its graph orientations in which every graph vertex has three inwardly directed and three outwardly directed edges. These are Eulerian orientations, with (6; 3)=20 possible local configurations at each graph vertex.

Baxter (1969) 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).

See also

Eulerian Orientation, Lieb's Square Ice Constant, Triangular Lattice

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References

Baxter, R. J. "F Model on a Triangular Lattice." J. Math. Phys. 10, 1211-1216, 1969. https://doi.org/10.1063/1.1664960.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."

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Twenty-Vertex Entropy Constant

Cite this as:

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

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