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 $⟨{\mathcal{N}}_{\mathrm{tot}}⟩$ 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 $⟨{\mathcal{N}}_{\mathrm{tot}}⟩\sim \exp (rL)$ with its length $L$. The growth rate $r$ is found to be directly related to the generalised Lyapunov exponent (GLE) which is a moment-generating function characterising 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.

v1: 6 pages main text + 14 pages supplemental material, 10 figures. v2: LaTeX, 79 pages, 18 eps figures, new material (Sections 6, 10, 11 & Appendices C, E, F, G)

References in corpus (14)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Topological and Dynamical Complexity of Random Neural Networks
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media
• Functionals of the Brownian motion, localization and metric graphs
• Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together
• Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices
• Universal distribution of threshold forces at the depinning transition
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Topology trivialization transition in random non-gradient autonomous ODE's on a sphere
• Landau levels with magnetic tunnelling in Weyl semimetal and magnetoconductance of ballistic $p$-$n$ junction
• Counting Vacua in Random Landscapes
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Computing the number of metastable states in infinite-range models

Cited by in corpus (20)

• Does scrambling equal chaos?
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Hessian spectrum at the global minimum of high-dimensional random landscapes
• Nonlinearity-generated Resilience in Large Complex Systems
• Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates
• Distribution of rare saddles in the $p$-spin energy landscape
• Manifolds in high dimensional random landscape: complexity of stationary points and depinning
• The number of saddles of the spherical $p$-spin model
• Kac-Rice fixed point analysis for single- and multi-layered complex systems
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• On-Site Potential Creates Complexity in Systems with Disordered Coupling
• The Distribution of Vacua in Random Landscape Potentials
• Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum
• Transmission of waves through a pinned elastic medium
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity
• Gelfand-Yaglom formula for functional determinants in higher dimensions
• An investigation of $\mathcal{P}\mathcal{T}$-symmetry breaking in tight-binding chains
• Disordered harmonic chain with random masses and springs: a combinatorial approach
• Representation theory and products of random matrices in $\text{SL}(2,\mathbb{R})$