# Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation: numerical study

Abstract : We study numerically the geometrical and free-energy fluctuations of a static one-dimensional (1D) interface with a short-range elasticity, submitted to a quenched random-bond Gaussian disorder of finite correlation length $\xi>0$, and at finite temperature $T$. Using the exact mapping from the static 1D interface to the 1+1 Directed Polymer (DP) growing in a continuous space, we focus our analysis on the disorder free-energy of the DP endpoint, a quantity which is strictly zero in absence of disorder and whose sample-to-sample fluctuations at a fixed growing 'time' $t$ inherit the statistical translation-invariance of the microscopic disorder explored by the DP. Constructing a new numerical scheme for the integration of the Kardar-Parisi-Zhang (KPZ) evolution equation obeyed by the free-energy, we address numerically the 'time'- and temperature-dependence of the disorder free-energy fluctuations at fixed finite $\xi$. We examine on one hand the amplitude $\tilde{D}_{t}$ and effective correlation length $\tilde{\xi}_t$ of the free-energy fluctuations, and on the other hand the imprint of the specific microscopic disorder correlator on the large-'time' shape of the free-energy two-point correlator. We observe numerically the crossover to a low-temperature regime below a finite characteristic temperature $T_c(\xi)$, as previously predicted by Gaussian-Variational-Method (GVM) computations and scaling arguments, and extensively investigated analytically in [Phys. Rev. E, 87 042406 (2013)]. Finally we address numerically the 'time'- and temperature-dependence of the roughness $B(t)$, which quantifies the DP endpoint transverse fluctuations, and we show how the amplitude $\tilde{D}_{\infty}(T,\xi)$ controls the different regimes experienced by $B(t)$ -- in agreement with the analytical predictions of a DP 'toymodel' approach.
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Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2013, 87 (6), pp.062405. <10.1103/PhysRevE.87.062405>
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https://hal.archives-ouvertes.fr/hal-00821806
Contributeur : Vivien Lecomte <>
Soumis le : lundi 13 mai 2013 - 10:24:39
Dernière modification le : mardi 11 octobre 2016 - 14:05:06

### Citation

Elisabeth Agoritsas, Vivien Lecomte, Thierry Giamarchi. Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation: numerical study. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2013, 87 (6), pp.062405. <10.1103/PhysRevE.87.062405>. <hal-00821806>

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