Finite-temperature and finite-time scaling of the directed polymer free energy with respect to its geometrical fluctuations

We study the fluctuations of the directed polymer in 1+1 dimensions in a Gaussian random environment with a finite correlation length $\xi$ and at finite temperature. We address the correspondence between the geometrical transverse fluctuations of the directed polymer, described by its roughness, and the fluctuations of its free energy, characterized by its two-point correlator. Analytical arguments are provided in favor of a generic scaling law between those quantities, at finite time, nonvanishing $\xi$, and explicit temperature dependence. Numerical results are in good agreement both for simulations on the discrete directed polymer and on a continuous directed polymer (with short-range correlated disorder). Applications to recent experiments on liquid crystals are discussed.

Figure 1

Schematic view of a continuous directed polymer of trajectory $y\left(t\right)$ starting in $y=0$ at time 0 and arriving in $y_1$ at time $t_1$, in a quenched random potential $V(t,y)$. These point-to-point configurations correspond to the “droplet geometry” of growth models.

Figure 2

Geometry of the discrete DP model we consider. Allowed steps are $y(t+1)-y\left(t\right)=\pm 1$. The longitudinal and transverse axes correspond to the directions $t$ and $y$, respectively.

Figure 3

Schematic plot of the correlator ${\overline{C}}_{\xi }(t,y)$ of $\overline{F}$, as a function of $y$ at fixed $t$. It is rounded close to the origin ($\left|y\right|\lesssim \xi$), varies as $\tilde{D}\left|y\right|$ for intermediate length ($\xi \lesssim \left|y\right|\lesssim {\ell }_t$), and goes to a constant at large length ($\left|y\right|\gtrsim {\ell }_t$).

Figure 4

Graph of the correlator ${\overline{C}}_{\xi }^{\text{lin}}(t,y)$ of $\overline{F}$ obtained in the linear approximation (C23), as a function of $y$ for different times $t$ (parameters are $c=1$, $D=1$, $T=1$, and $\xi =0.3$). Time increases geometrically from bottom ($t=2^{-4}$) to top ($t=2^{10}$) curves.

Figure 5

Bare free-energy fluctuations $\overline{C}(t,y)$ for the discrete DP at finite temperature $T=4$ and for two different times $t$. The late increase of $\overline{C}(t,y)$ for $t=512$ is a finite-size effect related to the solid-on-solid restriction of the DP model.

Figure 6

Rescaled free-energy fluctuations $\overline{C}(t,y)$ for the discrete DP at finite low temperature $T=0.5$ and various times, according to the scaling (37) with $\zeta ={\zeta }_{\mathrm{RM}}=\frac{2}{3}$. The inset shows the bare data $\overline{C}(t,y)$ against $y$ for timescales $t=4096,$ 8192, 16 384, 32 768 from bottom to top.

Figure 7

Rescaled free-energy fluctuations $\overline{C}(t,y)$ for the discrete DP at large times and high temperatures, according to the scaling (40). The inset shows the bare data $\overline{C}(t,y)$ against $y$ for timescales $t=4096,$ 8192, 16 384, 32 768 from bottom to top. Thin (thick) lines correspond to $T=4$ ($T=8$).

Figure 8

Rescaled free-energy fluctuations $\overline{C}(t,y)$ for the discrete DP at large times and low temperatures, according to the scaling (40). The inset shows the bare data $\overline{C}(t,y)$ against $y$ for timescales $t=4096,$ 8192, 16 384, 32 768 from bottom to top. Thin (thick) lines correspond to $T=0.5$ ($T=1$).

Figure 9

Rescaled free-energy fluctuations $\overline{C}(t,y)$ for the discrete DP at a fixed large time $t=16384$ and from low to high temperatures, according to the scaling (40). The inset shows the bare data $\overline{C}(t,y)$ against $y$. Temperatures are $T=0.5,1,2,4,8$ from top to bottom.

Figure 10

Full temperature-independent rescaled free-energy fluctuations $\overline{C}(t,y)$ for the discrete DP at a fixed large time, according to the scaling (41). Same data as in Fig. 9. The inset shows the parameter $\tilde{D}$ of (41) as a function of $T$.

Figure 11

Comparison between ${\widehat{C}}^1\left(y\right)$ defined in Eq. (41), extracted from the numerical data shown in Fig. 10 and $A{g}_{\mathrm{PS}}(y/B)$ (dashed line), where $g_{\mathrm{PS}}\left(y\right)$ is defined in Eq. (42) while $A$ and $B$ are two fitting parameters (determined to $A\simeq 4.08\pm 0.05$ and $B\simeq 2.07\pm 0.06$). The numerical evaluation of the exact expression for $g_{\mathrm{PS}}\left(y\right)$ was done in Refs. [76, 77].

Figure 12

Graph for the continuous DP of the rescaled correlator ${\overline{C}}^{\text{resc}}(t,\overline{y})$ defined in Eq. (45) as a function of $\overline{y}=y/\sqrt{B\left(t\right)}$ for different times $t$, at low temperature. Parameters are $c=1$, $D=8$, $T=0.4$, and $\xi =2$. Disorder average is performed over 1092 realizations. (a) Roughness $B\left(t\right)$ in log-log scale with the thermal [red line (bottom)] and random manifold [green line (top)] asymptotic regimes. (b) Corresponding bare correlator $\overline{C}(t,y)$. Time varies from 35 to 135 by steps of 25 from bottom to top curves.

Figure 13

Same graph as Fig. 12 but at higher temperature $T=1.5$ (disorder average is over 446 realizations).

Figure 14

Plot of the rescaled correlator ${\tilde{D}}^{-1}{\overline{C}}^{\text{resc}}(t,\overline{y})$ as a function of $\overline{y}$ at fixed large time $t$ and for different temperatures $T\in \{.4,.55,1.2,1.5,1.8\}$. (a) Corresponding original correlator $\overline{C}(t,y)$. Lighter color corresponds to lower temperature. (b) $\tilde{D}$ as a function of $T$ (up to a numerical factor) in log-log scale. Points are determined by least-square minimization (see text) with respect to a reference curve at $T=2$ which fixes $\tilde{D}{|}_{T=2}=1$. The line gives the expected slope of the large temperature asymptotics $\tilde{D}\propto \frac{1}{T}$.