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We consider various random models (directed polymer, ferromagnetic randomPotts model, Ising spin-glasses) in their disorder-dominated phases, where thefree-energy cost $F(L)$ of an excitation of length $L$ present fluctuationsthat grow as a power-law $\Delta F(L) \sim L^{\omega}$ with the so-calleddroplet 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 ofthis paper is to establish simple relations between these tail exponents$(\eta_-,\eta_+) $ and the droplet exponent $\omega$. We first prove theserelations for disordered models on diamond hierarchical lattices where exactrenormalizations exist for the probability distribution $\Pi(x)$. We theninterpret these relations via an analysis of the measure of the rare disorderconfigurations governing the tails. Our conclusion is that these relations,when expressed in terms of the dimensions of the bulk and of the excitationsurface are actually valid for general lattices.

Disorder-dominated phases of random systems : relations between tails exponents and scaling exponents