Manifolds in high dimensional random landscape: complexity of stationary points and depinning

We obtain explicit expressions for the annealed complexities associated respectively with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension $d<4$ embedded in a random medium of dimension $N \gg 1$ and confined by a parabolic potential with the curvature parameter $\mu$. These complexities are found to both vanish at the critical value $\mu_c$ identified as the Larkin mass. For $\mu<\mu_c$ the system is in complex phase corresponding to the replica symmetry breaking in its $T=0$ thermodynamics. The complexities vanish respectively quadratically (stationary points) and cubically (minima) at $\mu_c^-$. For $d\geq 1$ they admit a finite "massless" limit $\mu=0$ which is used to provide an upper bound for the depinning threshold under an applied force.