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.

20 pages, 16 figures. This preprint corresponds to the second part of the former arXiv:1209.0567v1, which has been split into two parts: arXiv:1209.0567v2 and the present preprint. Ancillary Files contain three animations of the time-evolution of the free-energy correlators $\bar{R}(t,y)$ and $\bar{C}(t,y)$, at low, intermediate and high temperature T (respectively T=0.4, T=1. and T=6.)

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Cited by in corpus (11)

• Numerical Approaches on Driven Elastic Interfaces in Random Media
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• Driven interfaces: from flow to creep through model reduction
• Effect of disorder geometry on the critical force in disordered elastic systems
• A numerical study of the statistics of roughness parameters for fluctuating interfaces
• "Sinc"-Noise for the KPZ Equation
• Nonsteady dynamic properties of a domain wall for the creep state under an alternating driving field
• Power countings versus physical scalings in disordered elastic systems - Case study of the one-dimensional interface
• Zero temperature limit for (1+1) directed polymers with correlated random potential
• Velocity distribution functions and intermittency in one-dimensional randomly forced Burgers turbulence
• Microscopic interplay of temperature and disorder of a one-dimensional elastic interface