Research output: Contribution to journal › Article

Yan V. Fyodorov, Pierre Le Doussal, Alberto Rosso, Christophe Texier

Original language | English |
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Pages (from-to) | 1-64 |

Journal | ANNALS OF PHYSICS |

Volume | 397 |

Early online date | 27 Jul 2018 |

DOIs | |

Publication status | Published - 2018 |

**Exponential number of equilibria_FYODOROV_Accepted24July2018_GREEN AAM (CC BY-NC-ND)**1_s2.0_S0003491618302008_main.pdf, 1.63 MB, application/pdf

27/07/2019

Accepted author manuscript

CC BY-NC-ND

© <2018> This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode

By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number 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 , grows exponentially with its length . The growth rate 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 is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate 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 , in the presence of an applied force, for elastic lines and -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.

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