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Hard Hexagon Entropy Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

A constant related to the Hard Square Entropy Constant. This constant is given by

\kappa_h\equiv \lim_{N\to\infty} [G(N)]^{1/N}=1.395485972\ldots,
\end{displaymath} (1)

where $G(N)$ is the number of configurations of nonattacking Kings on an $n\times n$ chessboard with regular hexagonal cells, where $N\equiv n^2$. Amazingly, $\kappa_h$ is algebraic and given by
\end{displaymath} (2)

$\displaystyle \kappa_1$ $\textstyle \equiv$ $\displaystyle 4^{-1} 3^{5/4} 11^{-5/12} c^{-2}$ (3)
$\displaystyle \kappa_2$ $\textstyle \equiv$ $\displaystyle [1-\sqrt{1-c}+\sqrt{2+c+2\sqrt{1+c+c^2}}\,]^2$ (4)
$\displaystyle \kappa_3$ $\textstyle \equiv$ $\displaystyle [-1-\sqrt{1-c}+\sqrt{2+c+2\sqrt{1+c+c^2}}\,]^2$ (5)
$\displaystyle \kappa_4$ $\textstyle \equiv$ $\displaystyle [\sqrt{1-a}+\sqrt{2+a+2\sqrt{1+a+a^2}}\,]^{-1/2}$ (6)
$\displaystyle a$ $\textstyle \equiv$ $\displaystyle -{\textstyle{124\over 363}} 11^{1/3}$ (7)
$\displaystyle b$ $\textstyle \equiv$ $\displaystyle {\textstyle{2501\over 11979}} 33^{1/2}$ (8)
$\displaystyle c$ $\textstyle \equiv$ $\displaystyle \{{\textstyle{1\over 4}}+{\textstyle{3\over 8}}a[(b+1)^{1/3}-(b-1)^{1/3}]\}^{1/3}.$ (9)

(Baxter 1980, Joyce 1988).


Baxter, R. J. ``Hard Hexagons: Exact Solution.'' J. Physics A 13, 1023-1030, 1980.

Finch, S. ``Favorite Mathematical Constants.''

Joyce, G. S. ``On the Hard Hexagon Model and the Theory of Modular Functions.'' Phil. Trans. Royal Soc. London A 325, 643-702, 1988.

Plouffe, S. ``Hard Hexagons Constant.''

© 1996-9 Eric W. Weisstein