Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature mu. We are interested in the mean spectral density rho(lambda) of the Hessian matrix K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for rho(lambda) for a fixed Ld in the N ->infinity limit extending d=0 results of our previous work (Fyodorov et al. in Ann Phys 397:1-64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting L ->infinity. In all cases we show that for a confinement curvature mu exceeding a critical value mu c, the so-called Larkin mass, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at mu ->mu c. For mu