Distribution of zeros in the rough geometry of fluctuating interfaces

We study numerically the correlations and the distribution of intervals between successive zeros in the fluctuating geometry of stochastic interfaces, described by the Edwards-Wilkinson equation. For equilibrium states we find that the distribution of interval lengths satisfies a truncated Sparre-Andersen theorem. We show that boundary-dependent finite-size effects induce nontrivial correlations, implying that the independent interval property is not exactly satisfied in finite systems. For out-of-equilibrium nonstationary states we derive the scaling law describing the temporal evolution of the density of zeros starting from an uncorrelated initial condition. As a by-product we derive a general criterion of the von Neumann's type to understand how discretization affects the stability of the numerical integration of stochastic interfaces. We consider both diffusive and spatially fractional dynamics. Our results provide an alternative experimental method for extracting universal information of fluctuating interfaces such as domain walls in thin ferromagnets or ferroelectrics, based exclusively on the detection of crossing points.

Figure 1

An interface with periodic boundary conditions and height $u_x\left(t\right)$ measured with respect to its average value $\overline{u}\left(t\right)$ defined on a lattice with $L$ sites. The zeros are the points where the interface intersects its mean value $\overline{u}\left(t\right)$ given by the crosses. Such zeros divide the lattice in $N$ intervals of lengths ${\ell }_i$, $i=1,\cdots ,N$. Since we work in discrete space, the detection of the zeros can be either selecting the site on the left of the point where the interface crosses its average value (filled circles at the bottom) or by choosing the site that is closer to the crossing point (empty circles). Notice how the choice of the method modifies the length of the intervals between consecutive zeros. In this case the second and the fourth gray zero are on the next site on the right if the nearest site was chosen. Moreover, when choosing a zero as the nearest site to the crossing point, one can find two zeros at the same site which in turn allows the existence of intervals of length $\ell =0$.

Figure 2

Normalized histogram of the distances between consecutive zeros $\ell$ for different sizes of the system compared to the modified Sparre-Andersen theorem $p\left(\ell \right)=\left(q\right(\ell )-q(\ell +1\left)\right)/Z$ with $Z$ the normalization factor given by Eq. (6) and $q\left(\ell \right)=\left(\frac{2\ell }{\ell }\right)2^{-2\ell }$ which is in good agreement even for small values of $\ell$. Vertical lines show the values of $L/2$ for $L=8192,16384,32768,65536,131072$ (in cyan, fuchsia, blue, green, and red, respectively, in the color version). The inset shows the tail of the distribution for $L$ large (which are the points for which Sparre-Andersen is no longer valid) with its points calculated from the average of the original data taken on logarithmic bins. In the limit $L\to \infty$ Sparre-Andersen is always satisfied. A scaling law is found for the probability of the intervals for large values of $\ell$ for which all the curves superpose. This law is $p(\ell ,L)\sim L^{-3/2}\widehat{p}(\ell /L)$ as explained in the text.

Figure 3

(a) Correlator of the noise ${\mathbb{K}}_{{\eta }_x}^{\mathrm{PBC}}$ of an interface with periodic boundary conditions using the exact Laplacian. (b) Correlator of the noise ${\mathbb{K}}_{{\eta }_x}^{\mathrm{PBC}}$ of an interface with periodic boundary conditions with the approximate Laplacian $(1-cosq)\approx \frac{1}{2}{q}^2$ [see Eq. (9)]. The increment correlations out of the diagonal persist even if we cut some Fourier modes.

Figure 4

Absolute value of the interval correlation $C\left(r\right)$ as defined in Eq. (11) for different sizes $L$ of the system and $r$ even. We show the rescaled correlation in log-normal scale which follows a scaling law $\left|C\left(r\right)\right|=L\left|\widehat{C}(r/L^{1/2})\right|$. The obtained scaling function $\widehat{C}$ is independent of $L$ (inset figure). $\left|C\right(r\left)\right|$ seems to follow a nontrivial law as compared to a pure exponential law $\sim e^{-3\widehat{r}/2}$ given by the solid line in the inset.

Figure 5

Stability of $S_q\left(t\right)$ as a function of time with $q=q^{*}=\pi$, $\mathrm{\Delta }t=1.1>\mathrm{\Delta }{t}_c$ and parameters $c=1$, $\zeta =1/2$. $S_{\pi }\left(t\right)$ is stable for all times for $a\le 1/2$ ($a=0,0.25,0.5$ in dark red, red and orange, respectively, from bottom up), however if $a$ becomes larger ($a=1,0.95,0.75$ in blue, green and cyan, respectively, from top down) $S_{\pi }\left(t\right)$ loses stability

Figure 6

Evolution of structure factor in time with discretization parameter $a=0$ in the evolution equation (12) with roughness exponent $\zeta =1/2$. The bending tail of $S_q\left(t\right)$ for $q$ close to $\pi$ is completely understood by means of the correction term proportional to $\mathrm{\Delta }t$ and the approximate Laplacian encoded in the variable $K_q=c\mathrm{\Delta }t{q}^2$ as derived in Eq. (19) (solid line). A dynamic scaling law is found to be $S_q=t^{-1}{\widehat{S}}_{q{t}^{1/2}}$ (inset).

Figure 7

Evolution of density of zeros in time for different system sizes: $L=2048,4096,8192$ (in red, green, and blue, respectively), averaged over 1000 realizations. There is a characteristic time $t_c\sim L^2$ from which the density reaches a steady state. The scaling law for the density of zeros is found to be $\rho (L,t)=L^{-1/2}\widehat{\rho }(t/L^2)$.