If n_1 and n_2 are even moderately large then we will have to resort to an approximation to E_{\pi}W . Consider the following Markov chain on \chi. Fix p \in (0, 1) and set X_1 = x_1 where x_1 \in \chi is an arbitrary proper configuration. Randomly choose a point (x, y) \in X and independently draw U \sim \mathrm{Uniform}(0, 1). If u \le p and all of the adjacent points are black then color (x, y) white leaving all other points alone. Otherwise, color (x, y) black and leave all other points alone. Call the resulting configuration X_1. Continuing in this fashion yields a Harris ergodic Markov chain \{X_1 , X_2 , X_3 , \ldots\} having \pi as its invariant distribution. It is now a simple matter to estimate E_{\pi} W with \overline{w_n}=\sum_{i=1}^{n} W(X_i)/n. Also, since \chi is finite (albeit potentially large) it is well known that X will converge exponentially fast to \pi which implies that a CLT holds for \overline{w_n}.
