Hard Square Entropy Constant

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

Let $F(n^2)$ be the number of binary $n\times n$ Matrices with no adjacent 1s (in either columns or rows). Define $N\equiv n^2$, then the hard square entropy constant is defined by

$$\kappa\equiv \lim_{N\to\infty} [F(N)]^{1/N}=1.503048082\ldots.$$

The quantity $\ln\kappa$ arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the Hard Hexagon Entropy Constant can also be defined.

References

Baxter, R. J.; Enting, I. G.; and Tsang, S. K. ``Hard-Square Lattice Gas.'' J. Statist. Phys. 22, 465-489, 1980.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/square/square.html

Pearce, P. A. and Seaton, K. A. ``A Classical Theory of Hard Squares.'' J. Statist. Phys. 53, 1061-1072, 1988.