On the critical region of long-range depinning transitions

The depinning transition of elastic interfaces with an elastic interaction kernel decaying as $1/r^{d+sigma}$ is characterized by critical exponents which continuously vary with $sigma$. These exponents are expected to be unique and universal, except in the fully coupled ($-d<sigmale 0$) limit, where they depend on the smooth or cuspy nature of the microscopic pinning potential. By accurately comparing the depinning transition for cuspy and smooth potentials in a specially devised depinning model, we explain such peculiar limit in terms of the vanishing of the critical region for smooth potentials, as we decrease $sigma$ from the short-range ($sigma geq 2$) to the fully coupled case. Our results have practical implications for the determination of critical depinning exponents and identification of depinning universality classes in concrete experimental depinning systems with non-local elasticity, such as contact lines of liquids and fractures.