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 ($-d<\sigma\le 0$) 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.