  We consider various random models (directed polymer, ferromagnetic random
Potts model, Ising spin-glasses) in their disorder-dominated phases, where the
free-energy cost $F(L)$ of an excitation of length $L$ present fluctuations
that grow as a power-law $\Delta F(L) \sim L^{\omega}$ with the so-called
droplet exponent $\omega>0$. We study the tails of the probability distribution
$\Pi(x)$ of the rescaled free-energy cost $x= \frac{F_L-
\bar{F_L}}{L^{\omega}}$, which are governed by two exponents $(\eta_-,\eta_+)$
defined by $\ln \Pi(x \to \pm \infty) \sim - | x |^{\eta_{\pm}}$. The aim of
this paper is to establish simple relations between these tail exponents
$(\eta_-,\eta_+) $ and the droplet exponent $\omega$. We first prove these
relations for disordered models on diamond hierarchical lattices where exact
renormalizations exist for the probability distribution $\Pi(x)$. We then
interpret these relations via an analysis of the measure of the rare disorder
configurations governing the tails. Our conclusion is that these relations,
when expressed in terms of the dimensions of the bulk and of the excitation
surface are actually valid for general lattices.
