Research output: Contribution to journal › Article

**Exponential number of equilibria and depinning threshold for a directed polymer in a random potential.** / Fyodorov, Yan V.; Le Doussal, Pierre; Rosso, Alberto; Texier, Christophe.

Research output: Contribution to journal › Article

Fyodorov, YV, Le Doussal, P, Rosso, A & Texier, C 2018, 'Exponential number of equilibria and depinning threshold for a directed polymer in a random potential', *ANNALS OF PHYSICS*, vol. 397, pp. 1-64. https://doi.org/10.1016/j.aop.2018.07.029

Fyodorov, Y. V., Le Doussal, P., Rosso, A., & Texier, C. (2018). Exponential number of equilibria and depinning threshold for a directed polymer in a random potential. *ANNALS OF PHYSICS*, *397*, 1-64. https://doi.org/10.1016/j.aop.2018.07.029

Fyodorov YV, Le Doussal P, Rosso A, Texier C. Exponential number of equilibria and depinning threshold for a directed polymer in a random potential. ANNALS OF PHYSICS. 2018;397:1-64. https://doi.org/10.1016/j.aop.2018.07.029

@article{c0596bf0df5c4f09b4b4a06eeb2c56d4,

title = "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 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 Schr{\"o}dinger 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.",

keywords = "Directed polymer in random medium, Pinning, Random Schr{\"o}dinger operator, Anderson localisation, Generalised Lyapunov exponent",

author = "Fyodorov, {Yan V.} and {Le Doussal}, Pierre and Alberto Rosso and Christophe Texier",

year = "2018",

doi = "10.1016/j.aop.2018.07.029",

language = "English",

volume = "397",

pages = "1--64",

journal = "ANNALS OF PHYSICS",

issn = "0003-4916",

publisher = "ACADEMIC PRESS INC",

}

TY - JOUR

T1 - Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

AU - Fyodorov, Yan V.

AU - Le Doussal, Pierre

AU - Rosso, Alberto

AU - Texier, Christophe

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Directed polymer in random medium

KW - Pinning

KW - Random Schrödinger operator

KW - Anderson localisation

KW - Generalised Lyapunov exponent

U2 - 10.1016/j.aop.2018.07.029

DO - 10.1016/j.aop.2018.07.029

M3 - Article

VL - 397

SP - 1

EP - 64

JO - ANNALS OF PHYSICS

JF - ANNALS OF PHYSICS

SN - 0003-4916

ER -

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