The zero temperature phase diagram of spin glasses on finite connectivity graphs is investigated, with or without magnetic field and/or ferromagnetic bias, for mean field (using the cavity method) and Edwards-Anderson (using numerical ground states computations) models. In the mean field case we show that the phase diagram is complex and compute the equivalent of the zero temperature de Almeida-Thouless line. In the 3d model however, we found a trivial phase diagram. This paper presents new analytical results as well as a rapid review of numerical works on ground states. Since its proposal in 1975, 1) the Edwards-Anderson (EA) model of spins glasses (SG) 2),3) — an Ising model with ferromagnetic and anti-ferromagnetic interactions — has been the subject of many studies and controversies. 4) A simple question such as the qualitative shape of the phase diagram is still matter of debates. 5) Here, we investigate the generic phase diagram of finite connectivity spin glasses at zero temperature, with magnetic field and/or ferromagnetic bias. It is motivated by the recent progresses in both numerical studies of T = 0 spin glasses (mainly obtained b
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