Research output: Contribution to journal › Article › peer-review

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

Original language | English |
---|---|

Pages (from-to) | 1-64 |

Journal | ANNALS OF PHYSICS |

Volume | 397 |

Early online date | 27 Jul 2018 |

DOIs | |

Accepted/In press | 24 Jul 2018 |

E-pub ahead of print | 27 Jul 2018 |

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

Uploaded date:02 Aug 2018

Version:Accepted author manuscript

Licence: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.

No data available

King's College London - Homepage

© 2020 King's College London | Strand | London WC2R 2LS | England | United Kingdom | Tel +44 (0)20 7836 5454