Non-equilibrium dynamics of polymers and interfaces in random media : conjecture $\psi=d_s/2$ for the barrier exponent

We consider various random models (directed polymer, random ferromagnets, spin-glasses) in their disorder-dominated phases, where the free-energy cost $F(L)$ of an excitation of length $L$ presents fluctuations that grow as a power-law $\Delta F(L) \sim L^{\theta}$ with the 'droplet' exponent $\theta$. Within the droplet theory, the energy and entropy of such excitations present fluctuations that grow as $\Delta E(L) \sim \Delta S(L) \sim L^{d_s/2}$ where $d_s$ is the dimension of the surface of the excitation. These systems usually present a positive 'chaos' exponent $\zeta=d_s/2-\theta>0$, meaning that the free-energy fluctuation of order $L^{\theta}$ is a near-cancellation of much bigger energy and entropy fluctuations of order $L^{d_s/2}$. Within the standard droplet theory, the dynamics is characterized by a barrier exponent $\psi$ satisfying the bounds $\theta \leq \psi \leq d-1$. In this paper, we argue that a natural value for this barrier exponent is $\psi=d_s/2$ : (i) for the directed polymer where $d_s=1$, this corresponds to $\psi=1/2$ in all dimensions; (ii) for disordered ferromagnets where $d_s=d-1$, this corresponds to $\psi=(d-1)/2$; (iii) for spin-glasses where interfaces have a non-trivial dimension $d_s$ known numerically, our conjecture $\psi=d_s/2$ gives numerical predictions in $d=2$ and $d=3$. We compare these values with the available numerical results for each case, in particular with the measure $\psi \simeq 0.49$ of Kolton, Rosso, Giamarchi, Phys. Rev. Lett. 95, 180604 (2005) for the non-equilibrium dynamics of a directed elastic string.