Let C_n be the origin-containing cluster in subcritical percolation on the
lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact,
connected, origin-containing subsets of R^d, endowed with the Hausdorff metric
delta. When d >= 2, and Gamma is any open subset of Omega, we prove: lim_{n \to
\infty}(1/n) \log P(C_n \in \Gamma) = -\inf_{S \in \Gamma} \lambda(S) where
lambda(S) is the one-dimensional Hausdorff measure of S defined using the {\em
correlation norm}: ||u|| := \lim_{n \to \infty} - \frac{1}{n} \log P (u_n \in
C_n) where u_n is u rounded to the nearest element of (1/n)Z^d. Given points
a^1, >..., a^k in R^d, there are finitely many correlation-norm Steiner trees
spanning these points and the origin. We show that if the C_n are each
conditioned to contain the points a^1_n,..., a^k_n, then the probability that
C_n fails to approximate one of these trees decays exponentially in n.