Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation: Numerical study

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 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 $\xi$. 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 $T_c\left(\xi \right)$, 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 $B\left(t\right)$, which quantifies the DP end point transverse fluctuations, and we show how the amplitude ${\tilde{D}}_{\infty }(T,\xi )$ controls the different regimes experienced by $B\left(t\right)$—in agreement with the analytical predictions of a DP toy model approach.

Figure 1

The paths represent different realizations of a static 1D interface of length $t_1$ in the 1+1 DP representation: One polymer extremity is attached at a fixed origin $(0,0)$ while its end point at time $t_1$ is fluctuating. In a given disorder realization $V(t,y)$, the distribution of the end point $y_1=y\left(t_1\right)$ is denoted by ${\mathcal{P}}_V(t_1,y_1)$ and its mean variance at time $t_1$ defines the DP “roughness” $B\left(t_1\right)$. The DP description of the interface as growing along a “time” direction allows us to use tools of stochastic processes and amounts, in the language of the 1D interface, to study the effective statistics of the DP end point at a fixed length scale $t_1$, having integrated the fluctuations at shorter length scales.

Figure 2

The finite-time correlator ${\overline{R}}^{\text{lin}}(t,y)$ (thin purple lines) for the CubicS disorder correlator $R_{\xi }\left(y\right)=R_{\xi }^{\text{CubicS}}\left(y\right)$ defined in (A2) and (A3) and plotted as a function of $y$ for different times and compared to its infinite-time limit $R_{\xi }\left(y\right)$ (thick red line). The central peak develops with increasing times from the flat initial condition $\overline{R}(0,y)\equiv 0$. Parameters are $\xi =2$, $c=1$, $D=1$, and $T=1$. Left inset: Same behavior for ${\overline{C}}^{\text{lin}}(t,y)$. Right inset: Same behavior for $\frac{1}{2}{\overline{C}}^{\text{lin}}(t,y)={\int }_0^{y}d{y}^{\prime }{\overline{R}}^{\text{lin}}(t,y^{\prime })$.

Figure 3

Illustration of the numerical procedure for the 1+1 continuous DP in the finite box $\left[0,t_m\right]\times \left[-y_m,y_m\right]$ with the initial condition at $t_0=0.1$ taken as purely thermal $\mathcal{P}(t_0,y)=Z_{V\equiv 0}(t_0,y)$ (9) (red curve). There are three levels of discretization grids by decreasing scale: the grid for the random potential ${\xi }_t^{\text{grid}}=1$ and ${\xi }_y^{\text{grid}}=2$ (the gray box on the left side), the linear grids for the recording of data $\Delta t^{\text{lin}}=t_m/80$ and $\Delta y^{\text{lin}}=2y_m/100$, and the logarithmic grid $t_j^{\text{log}}=t_0{(t_m/t_0)}^{j/80}$ with $j=0\cdots 80$. The microscopic grid for the numerical integration is adapted in both the $(t,y)$ directions by use of mathematica. At a given configuration of disorder $V(t,y)$ as illustrated in Fig. 10, the PDF ${\mathcal{P}}_V(t,y)$ is computed at increasing time following (14), with the boundary condition ${\partial }_y{\overline{F}}_V(t,\pm y_m)=0$ and the ad hoc normalization (34), as illustrated at $t_0<t_1<t_2$ (blue curves). See Fig. 1 for the interpretation in terms of DP trajectories.

Figure 4

Effective disorder correlator $\overline{R}(t,y)$ measured numerically at fixed temperature $T\in \left\{0.35,1,6\right\}$. The different times $t\in \left[0.1,40\right]$ are separated into two subsets $\left[0,t_{\text{min}}\right]$ (left) and $\left[t_{\text{min}},40\right]$ (right) with $t_{\text{min}}=25$, with a time step $\Delta t=1$. Left side ($y\in \left[-10,10\right]$): Initial development of the central peak (as indicated by the arrows), saturation, and accumulation of the curves in the vicinity of $y=0$. Right side ($y\in \left[-40,40\right]$): Large times correlators and their average over $t>t_{\text{min}}$ (superimposed black curve). Center: Saturation correlator ${\overline{R}}_{\text{sat}}\left(y\right)$ (black dots) with its Gaussian (35) (red, positive) and SincG (36) (blue, with negative oscillations) fitting functions, CubicS collapsing exactly on SincG; see Table in Appendix  for the explicit values of the fitting parameters.

Figure 5

Temperature-dependent amplitude ${\tilde{D}}_t$ and typical spread ${\tilde{\xi }}_t$ for SincG for $T\in \left\{0.35,1,2,3,4,6\right\}$ as listed in Appendix . See Appendix 10a for the error bars of the minimum and maximum temperatures. For increasing temperatures (blue to red), ${\tilde{D}}_t$ decreases, whereas ${\tilde{\xi }}_t$ slightly increases (${\xi }_y^{\text{grid}}=2$ for reference), resulting in an overall damping of the correlator ${\overline{R}}_{\text{sat}}\left(y\right)$ (see Fig. 6).

Figure 6

Temperature dependence of the maximum of ${\overline{R}}_{\text{sat}}\left(y\right)$ over times $t\in \left[25,40\right]$, measured by three different ways (from bottom to top): first, deduced by the SincG and Gauss fits as in Fig. 13 (respectively, blue and red dots); second, measured numerically directly ${\overline{R}}_{\text{sat}}\left(0\right)$ (black crosses); and, third, measured indirectly from the linear slope of $-2c\overline{{\overline{F}}_V(t,y)}$ in Fig. 16 (purple dots) and systematically slightly overestimated. Upper inset: Saturation correlator ${\overline{R}}_{\text{sat}}\left(y\right)$ for temperatures $T\in \left\{0.35,1,2,3,4,6\right\}$ (as indicated by the arrow) and $t_{\text{min}}=25$ as listed in Appendix . Lower inset: The excellent linear behavior of $\overline{{\overline{F}}_V(t,y)}$ yields vanishing error bars for the slope on the range $t\in \left[25,40\right]$.

Figure 7

Roughness $B\left(t\right)=\overline{\langle y{\left(t\right)}^2\rangle }$ and its disorder component $B_{\text{dis}}\left(t\right)=B\left(t\right)-\frac{Tt}{c}={\overline{\langle y{\left(t\right)}^2\rangle }}^c$ measured numerically at fixed temperature $T\in \{0.35,1,1.8\}$. Main: Focus on the crossover of $B\left(t\right)$ (thick) from the pure thermal behavior $B_{\text{th}}\left(t\right)=\frac{Tt}{c}$ (dotted red) at short times to the asymptotic $B_{\text{RM}}\left(t\right)\sim {\tilde{D}}_{\infty }{\left(T\right)}^{2/3}{t}^{4/3}$ (dotted green) at asymptotically large times; $B_{\text{dis}}\left(t\right)$ is the thinner lower curve. Inset: Focus on the short-times crossover of $B_{\text{dis}}\left(t\right)$ (thick) in parallel to $B\left(t\right)$ (thin).

Figure 8

Temperature-dependent roughness $B\left(t\right)=\overline{\langle y{\left(t\right)}^2\rangle }$ and its disorder component $B_{\text{dis}}\left(t\right)$ (in logarithmic scale, respectively, the top and bottom curves) for $T\in [0.35,6]$ as listed in Appendix  (blue to red for increasing temperatures, as indicated by the black arrows). $B_{\text{dis}}\left(t\right)$ for $T=6$ is indicated in dotted red. See Fig. 7 for the description of the different regimes for three fixed temperatures.

Figure 9

Logarithmic slope of the roughness, $\zeta \left(t\right)$ and ${\zeta }_{\text{dis}}\left(t\right)$, obtained, respectively, from $B\left(t\right)$ and $B_{\text{dis}}\left(t\right)$ of Fig. 8 for $T\in \left[0.35,1\right]$ (blue to red for increasing temperatures, as indicated by the black arrow) and $T=1.8$ (in thin dashed dotted black). The thermal exponent ${\zeta }_{\text{th}}={\zeta }_{\text{EW}}=\frac{1}{2}$ and the RM exponent ${\zeta }_{\text{RM}}={\zeta }_{\text{KPZ}}=\frac{2}{3}$ are indicated by the thick dash-dotted lines, respectively red and green. At larger times those logarithmic slopes are very noisy but still oscillate around the expected value ${\zeta }_{\text{RM}}=\frac{2}{3}$.

Figure 10

Top: Visualization of a single disorder configuration $V(t,y)$; the mesh accounts for the grid $\left\{y_j\right\}$ and fluctuates smoothly. Bottom: Graph of the numerically measured effective 1D cubic-splined disorder correlator (blue), with the SincG and CubicS (red, respectively dashed and continuous) superimposed with a slight translation for more visibility. Inset: Full effective disorder correlator at fixed $t=0$ computed over $10000$ disorder configurations with $D^{\text{grid}}=4$, ${\xi }_y^{\text{grid}}=2$, and ${\xi }_t^{\text{grid}}=1$.

Figure 11

Fitting parameter ${\tilde{D}}_t$ at fixed low versus high temperature $T\in \left\{0.35,6\right\}$ with respect to Gauss (red, top) and SincG (blue, bottom), CubicS collapsing exactly on SincG. The straight lines indicate the respective average value ${\tilde{D}}_{\infty }$ for $t>t_{\text{min}}=25$, whose standard deviation is given explicitly for each fit.

Figure 12

Fitting parameter ${\tilde{\xi }}_t$ at fixed low versus high temperature $T\in \left\{0.35,6\right\}$, computed conjointly to ${\tilde{D}}_t$ of Fig. 11 and following the same convention regarding the legend with respect to Gauss (red, bottom) and SincG (blue, top).

Figure 13

Top: Time evolution of the maximum of the correlator peak $\overline{R}(t,0)$ at fixed temperature $T\in \left\{0.35,1,6\right\}$ measured numerically (black dots) or predicted by the fitting functions via ${\overline{R}}^{\text{fit}}(t,0)={\mathcal{R}}_{\tilde{\xi }=1}^{\text{fit}}\left(0\right){\tilde{D}}_t^{\text{fit}}/{\tilde{\xi }}_t^{\text{fit}}$ $\{$(top) red for Gauss (35) and blue for SincG (36); (bottom) cyan for CubicS [(A2) and (A3)]$\}$. The curves for SincG and CubicS systematically coincide. Bottom left: Close-up of the high-$T$ case, where the SincG and CubicS predict more accurately the peak at high $T$, whereas at low $T$ Gauss seems more suited. Bottom right: Decrease in the relative differences between the Gauss and SincG parameters with increasing temperature, averaged on the large-time window $t\in \left[25,40\right]$: $\frac{\Delta {\tilde{D}}_{\infty }}{{\tilde{D}}_{\infty }}=\frac{{\tilde{D}}_{\infty }^{\text{SincG}}-{\tilde{D}}_{\infty }^{\text{Gauss}}}{{\tilde{D}}_{\infty }^{\text{SincG}}}$, $\frac{\Delta {\tilde{\xi }}_{\infty }}{{\tilde{\xi }}_{\infty }}=\frac{{\tilde{\xi }}_{\infty }^{\text{SincG}}-{\tilde{\xi }}_{\infty }^{\text{Gauss}}{\xi }^{\text{SincG}}/{\xi }^{\text{Gauss}}}{{\tilde{\xi }}_{\infty }^{\text{SincG}}}$ (see the end of Appendix ), and $\frac{\Delta {\overline{R}}_{\text{sat}}\left(0\right)}{{\overline{R}}_{\text{sat}}}=\frac{{\overline{R}}_{\text{sat}}^{\text{SincG}}\left(0\right)-{\overline{R}}_{\text{sat}}^{\text{Gauss}}\left(0\right)}{{\overline{R}}_{\text{sat}}^{\text{SincG}}\left(0\right)}$.

Figure 14

Temperature dependence of ${\tilde{D}}_{\infty }$ with respect to Gauss (red, top) and SincG (blue, bottom). The straight line $cD/T$ indicates the expected behavior in the high-$T$ regime ($cD=8$ for our data). Inset: Corresponding mean and standard deviation for the average of ${\tilde{D}}_t$ over $t>t_{\text{min}}=25$.

Figure 15

Top: Temperature dependence of ${\tilde{\xi }}_{\infty }$ with respect to Gauss (red, bottom) and SincG (blue, top), normalized for Gauss with ${\xi }^{\text{SincG}}/{\xi }^{\text{Gauss}}\approx 3.6$ (cf. Appendix ). Bottom: Corresponding mean and standard deviation for the average of ${\tilde{\xi }}_t$ over $t>t_{\text{min}}=25$.

Figure 16

Disorder average of ${\overline{F}}_V(t,y)$ at fixed temperature $T\in \left\{0.35,1,6\right\}$. Top: $\overline{{\overline{F}}_V(t,y)}$ as a function of $y$ for increasing times $t\in \left[0.1,40\right]$ with time steps $\Delta t=2.5$ (top to bottom). Bottom: $-2c\overline{{\overline{F}}_V(t,y)}$ averaged over $y$ as a function time, the error bars indicating the corresponding standard deviation; the result of the linear fit for $t>t_{\text{min}}=25$ is indicated by the straight lines in black. Inset: Zoom on the short-time behavior in a logarithmic scale, which shows qualitatively the existence of a saturation time $t_{\text{sat}}\lesssim 10$ marking the beginning of the linear regime at large times (the dotted lines guide the eye for a quadratic behavior $\sim t^2$ as predicted analytically in Ref. [42]).