The order-parameter susceptibility χ of dilute Ising models with random fields and dilute antiferromagnets in a uniform field are studied for low temperatures and fields with use of low-concentration expansions, scaling theories, and exact solutions on the Cayley tree to elucidate the behavior near the percolation threshold at concentration pc. On the Cayley tree, as well as for d>6, both models have a zero-temperature susceptibility which diverges as |ln(pc-p)|. For spatial dimensions 1 6.