Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

Using the Kac-Rice approach, we show that the mean number 〈Ntot〉 of all possible equilibria of an elastic line (directed polymer), confined in an harmonic well and submitted to a quenched random Gaussian potential, grows exponentially 〈Ntot〉 ∼ exp (r L) with its length L. The growth rate r is found to be directly related to the fluctuations of the Lyapunov exponent of an associated Anderson localization problem of a 1d Schrödinger equation in a random potential. For strong confinement, the rate r is small and given by a non-perturbative (instanton) contribution to the Lyapunov exponent. For weak confinement, the rate r is found proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape ”topology trivialization” phenomenon, we obtain an upper bound for the depinning threshold fc, in presence of an applied force.