Exact ground states of three-dimensional random field Ising magnets (RFIM) with Gaussian distribution of the disorder are calculated using graph-theoretical algorithms. Systems for different strengths $h$ of the random fields and sizes up to $N=96^3$ are considered. By numerically differentiating the bond-energy with respect to $h$ a specific-heat like quantity is obtained, which does not appear to diverge at the critical point but rather exhibits a cusp. We also consider the effect of a small uniform magnetic field, which allows us to calculate the $T=0$ susceptibility. From a finite-size scaling analysis, we obtain the critical exponents $\nu=1.32(7)$, $\alpha=-0.63(7)$, $\eta=0.50(3)$ and find that the critical strength of the random field is $h_c=2.28(1)$. We discuss the significance of the result that $\alpha$ appears to be strongly negative.
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