  We consider strongly coupled field theories at large N on black hole
backgrounds. At sufficiently high Hawking temperature T_H, one expects a phase
where the black hole is in equilibrium with a deconfined plasma. We explore
this phase in the context of the AdS/CFT correspondence, and argue that two
possible behaviors may result. At a given Hawking temperature T_H, field
theories on large black hole backgrounds will couple strongly to the plasma and
will be dual to novel bulk spacetimes having a single connected horizon which
we dub {\it black funnels}. We construct examples of black funnels in low
spacetime dimensions for different classes of field theory black holes. In this
case, perturbing the equilibrium state results in the field theory exchanging
heat with the black hole at a rate typical of conduction through deconfined
plasma. In contrast, we argue that due to the finite physical size of plasma
excitations, smaller black holes will couple only weakly to the field theory
excitations. This situation is dual to bulk solutions containing two
disconnected horizons which remain to be constructed. Here perturbations lead
to heat exchange at a level typical of confined phases, even when T_H remains
far above any deconfinement transition. At least at large N and strong
coupling, these two behaviors are separated by a sharp transition. Our results
also suggest a richer class of brane-world black holes than hitherto
anticipated.
