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Hard Hexagon Entropy Constant

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The hard hexagon entropy constant is the growth constant for configurations of nonattacking kings on an n×n chessboard with regular hexagonal cells as n tends to infinity.

Use an n×n array of 0s and 1s with entries a_(r+|_j/2_|,j) for 1<=r,j<=n, where |_x_| is the floor function. For n=4, the array is

[a_(11) a_(23) ; a_(22) a_(34); a_(21) a_(33) ; a_(32) a_(44); a_(31) a_(43) ; a_(42) a_(54); a_(41) a_(53) ; a_(52) a_(64)].
(1)

Call two elements a_(ij) adjacent if they lie in positions (i,j) and (i+1,j), (i,j) and (i,j+1), or (i,j) and (i+1,j+1) for some i,j. Call G(n) the number of such arrays with no pairs of adjacent 1s.

The first few values of G(n) for n=1, 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).

The constant is given by

kappa_h=lim_(n->infty)[G(n)]^(1/n^2)
(2)
=1.395485972...
(3)

(OEIS A085851).

Amazingly, kappa_h is algebraic and is given by

kappa_h=kappa_1kappa_2kappa_3kappa_4,
(4)

where

kappa_1=4^(-1)3^(5/4)11^(-5/12)c^(-2)
(5)
kappa_2=[1-sqrt(1-c)+sqrt(2+c+2sqrt(1+c+c^2))]^2
(6)
kappa_3=[-1-sqrt(1-c)+sqrt(2+c+2sqrt(1+c+c^2))]^2
(7)
kappa_4=[sqrt(1-a)+sqrt(2+a+2sqrt(1+a+a^2))]^(-1/2),
(8)

with a=-(124)/(363)11^(1/3), b=(2501)/(11979)33^(1/2), and c={1/4+3/8a[(b+1)^(1/3)-(b-1)^(1/3)]}^(1/3) (Baxter 1980, Joyce 1988ab).

The variable c can be expressed in terms of the tribonacci constant

t=(x^3-x^2-x-1)_1,
(9)

where (P(x))_n denotes the nth polynomial root of P(x) in the ordering of the Wolfram Language. The equivalent expressions are

c=[1/4-(31(13t+81))/(242(32t+7))]^(1/3)
(10)
=[1/4-(31(32t^2-39t-19))/(2662)]^(1/3)
(11)
=(10307264x^9-7730448x^6+3839236x^3-161051)_1
(12)

(T. Piezas III, pers. comm., Feb. 11, 2006).

Explicitly, kappa_h is the unique positive polynomial root

kappa_h=(25937424601z^(24)+2013290651222784z^(22)+2505062311720673792z^(20)+797726698866658379776z^(18)+7449488310131083100160z^(16)+2958015038376958230528z^(14)-72405670285649161617408z^(12)+107155448150443388043264z^(10)-71220809441400405884928z^8-73347491183630103871488z^6+97143135277377575190528z^4-32751691810479015985152)_2.
(13)

See also

Hard Square Entropy Constant, Kings Problem, Tribonacci Constant

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References

Baxter, R. J. "Hard Hexagons: Exact Solution." J. Phys. A: Math. Gen. 13, L61-L70, 1980. https://doi.org/10.1088/0305-4470/13/3/007.Baxter, R. J. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.Finch, S. R. "Hard Square Entropy Constant." §5.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 342-349, 2003.Joyce, G. S. "On the Hard-Hexagon Model and the Theory of Modular Functions." Phil. Trans. Royal Soc. London A 325, 643-702, 1988a. https://doi.org/10.1098/rsta.1988.0077.Joyce, G. S. "Exact Results for the Activity and Isothermal Compressibility of the Hard-Hexagon Model." J. Phys. A: Math. Gen. 21, L983-L988, 1988b. https://doi.org/10.1088/0305-4470/21/20/005.Katzenelson, J. and Kurshan, R. P. "S/R: A Language for Specifying Protocols and Other Coordinating Processes." In Proc. IEEE Conf. Comput. Comm., pp. 286-292, 1986.Sloane, N. J. A. Sequences A066863 and A085851 in "The On-Line Encyclopedia of Integer Sequences."

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Hard Hexagon Entropy Constant

Cite this as:

Weisstein, Eric W. "Hard Hexagon Entropy Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HardHexagonEntropyConstant.html

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