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

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.