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< dpc- p)−(γp−βp)/2, where γp and βp are percolation exponents associated with the susceptibility and order parameter. At d=6, the susceptibilities diverge as |ln(pc-p)|9/7. For d=1, exact results show that the two models have different critical exponents at the percolation threshold. The (finite-length) series at d=2 seems to exhibit different critical exponents for the two models. At p=pc, the susceptibilities diverge in the limit of zero field h as χ~h-(γp-βp)/(γp+βp) for d9/7 for d=6, and as χ~|lnh| for d>6.