The number of independent sets in a grid graph

If f(m; n) is the (vertex) independence number of the m x n grid graph, then we show that the double limit eta [GRAPHICS] lim(m,n-->infinity);n ! f(m; n) 1/mn exists, thereby refining earlier results of Weber [Rostock. Math. Kolloq., 34 (1988), pp. 28-36] and Engel [Fibonacci Quart., 28 (1990), pp. 72-78]. We establish upper and lower bounds for eta and prove that 1.503047782... is less-than-or-equal-to eta less-than-or-equal to 1.5035148.... Numerical computations suggest that the true value of eta (the "hard square constant") is around 1.5030480824753323....