  With respect to usual thermal ferromagnetic transitions, the zero-temperature
finite-disorder critical point of the Random-field Ising model (RFIM) has the
peculiarity to involve some 'droplet' exponent $\theta$ that enters the
generalized hyperscaling relation $2-\alpha= \nu (d-\theta)$. In the present
paper, to better understand the meaning of this droplet exponent $\theta$
beyond its role in the thermodynamics, we discuss the statistics of low-energy
excitations generated by an imposed single spin-flip with respect to the ground
state, as well as the statistics of equilibrium avalanches i.e. the
magnetization jumps that occur in the sequence of ground-states as a function
of the external magnetic field. The droplet scaling theory predicts that the
distribution $dl/l^{1+\theta}$ of the linear-size $l$ of low-energy excitations
transforms into the distribution $ds/s^{1+{\theta/d_f}}$ for the size $s$
(number of spins) of excitations of fractal dimension $d_f$ ($s \sim l^{d_f}$).
In the non-mean-field region $d<d_c$, droplets are compact $d_f=d$, whereas in
the mean-field region $d>d_c$, droplets have a fractal dimension $d_f=2 \theta$
leading to the well-known mean-field result $ds/s^{3/2}$. Zero-field
equilibrium avalanches are expected to display the same distribution
$ds/s^{1+{\theta/d_f}}$. We also discuss the statistics of equilibrium
avalanches integrated over the external field and finite-size behaviors. These
expectations are checked numerically for the Dyson hierarchical version of the
RFIM, where the droplet exponent $\theta(\sigma)$ can be varied as a function
of the effective long-range interaction $J(r) \sim 1/r^{d+\sigma}$ in $d=1$.
