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

Abstract

By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number N tot of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimension d = 1 + 1, grows exponentially N tot ∼ exp (r L) with its length L. The growth rate r is found to be directly related to the generalized Lyapunov exponent (GLE) which is a momentgenerating function characterizing the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rate r is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate r is found to be 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 f c , in the presence of an applied force, for elastic lines and d-dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.

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

“…The manifold is usually parameterized by a N -component real displacement field u(x) ∈ R N , where x belongs to an internal space x ∈ Ω. Ω can be either a finite collection of points, such as a subset L d of an internal space of dimension d, Ω ⊂ Z d , for discrete models, or Ω ⊂ R d in a continous setting. The case d = 1 corresponds to a line in N dimensions and for N = 1 was studied in the present context in [10]. The case d = 0 usually refers below to Ω being a single point, previously studied in [11] in the large N limit, and the present study can be seen as its generalization to a manifold.…”

“…While these results predict large scale properties of the low energy configurations, little is known about the detailed statistical structure of the complex energy landscape of pinned manifolds. This relates to the broad effort of understanding the statistical structure of stationary points (minima, maxima and saddles) of random landscapes which is of steady interest in theoretical physics [30][31][32][33][34][35][36][37][38][39], with recent applications to statistical physics [10,[34][35][36][38][39][40][41], neural networks and complex dynamics [42][43][44][45][46], string theory [47,48] and cosmology [49,50]. It is also of active current interest in pure and applied mathematics [51][52][53][54][55][56][57][58][59][60], For the model (1)- (2) in the simplest case d = 0 (x is a single point), the mean number of stationary points and of minima of the energy function was investigated in the limit of large N 1 in [35,38,39], see also [37,…”

“…As has been already mentioned, with setting µ ef f = µ the above expressions (10,11) provides the mean resolvent and the mean spectral density ρ(λ) for the manifold Hessian around a generic point of the disordered landscape. Such object is interesting by itself and we study the shapes of the spectral density in detail for several examples.…”

Kac–Rice fixed point analysis for single- and multi-layered complex systems

“…This type of transition from a 'trivial' landscape to 'complex' landscape has been observed in large number of models over the recent years, see e.g. [12,33,17,14,16,18]. It has been suggested to refer to such transitions as topological trivialisation [15,16,28,14].…”

Persistent stability of a chaotic system

“…We remark that Fyodorov et al have studied very closely related equations which occur in modelling pinning of polymers, including a related WKB analysis [21].…”