Abstract: The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength-dependent length scale. This breakup length scale L{sub b} scales exponentially with the squared random field, exp(A/{Delta}{sup 2}). By adding an external field H, we then study the susceptibility in the ground state. If L{gt}L{sub b}, domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as L{sup {minus}2}. We define a random field strength-dependent critical external field value {+-}H{sub c}({Delta}) for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing {Delta} and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) H{sub c}{similar_to}({Delta}{minus}{Delta}{sub c}){sup {delta}}, where {Delta}{sub c}=1.65{+-}0.05 and {delta}=2.05{+-}0.10. Below {Delta}{sub c} the systems should percolate even when H=0. This implies that even for H=0 above L{sub b} the domains can be fractal at low random fields, suchmore » that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation D{sub f}=91/48. The structure of the spanning clusters is studied by defining red clusters, in analogy to the {open_quotes}red sites{close_quotes} of ordinary site percolation. The sizes of red clusters define an extra length scale, independent of L.« less