Static fluctuations of a thick one-dimensional 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 ξ>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 end point, a quantity which is strictly zero in the 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 evolution equation obeyed by the free energy, we address numerically the time and temperature dependence of the disorder free-energy fluctuations at fixed finite ξ. We examine, on one hand, the amplitude D[over ̃](t) and effective correlation length ξ[over ̃](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)(ξ), as previously predicted by Gaussian variational method 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 end point transverse fluctuations, and we show how the amplitude D[over ̃](∞)(T,ξ) controls the different regimes experienced by B(t)-in agreement with the analytical predictions of a DP toy model approach.