Figure 1

Multifractal exponents $D_q$ vs $q$ for (top) $\gamma =0.05$, (middle) $\gamma =0.5$, and (bottom) $\gamma =0.95$. Red solid curves: WP at $t=100$ (including envelope and fluctuation effects; see text); blue dotted curves: fluctuations corresponding to the WP divided by the average WP over all realizations; green dashed curves: fluctuations corresponding to the WP divided by a smoothed average WP; purple dot-dashed curves: fluctuations corresponding to the WP divided by the analytical average WP Eq. (14) (see text). In the top and middle panels the three latter curves are mostly indistinguishable. Shaded areas indicate standard error in the least-squares fit. The insets show examples of fit for the WP and $q=2$. The $D_q$ have been extracted from 1000 random phase realizations of size  $N=2^{16}$ by the BC method applied on the $2^{15}$ central components with box sizes ranging from 1 to 64. All quantities on the figure are dimensionless.Reuse & Permissions

Figure 2

Multifractal exponents $D_q$ vs $q$ for (top) $\gamma =0.05$, (middle) $\gamma =0.5$, and (bottom) $\gamma =0.95$. Red solid curves: WP at $t=100$; blue dotted curves: $t=\infty$ limit; green dashed curves: eigenvectors of (1); black dot-dashed curves: analytical theory for eigenvectors (see text). Shaded areas indicate standard error in the least-squares fit. The insets show the average WP (initially localized at $P=0$; dotted yellow curve) and analytical formula (14) (black solid curve). The $D_q$ of WP have been extracted from 1000 random phase realizations of size $N=2^{16}$ by the BC method applied on the $2^{15}$ central components with box sizes from 1 to 64. The $D_q$ in the $t=\infty$ limit and for eigenvectors have been extracted from $2^{13}$ vectors of size $2^{12}$ by the BC method with box sizes from 8 to 512. All quantities on the figure are dimensionless.Reuse & Permissions

Figure 3

Density plot of $D_1$ as a function of $\gamma$ and time $t$. Colors denote multifractality strength from white (strong) to red or gray (weak). The exponents $D_1$ have been extracted from 100 random phase realizations of size $N=2^{13}$ by the BC method with box sizes from 1 to 64. All quantities on the figure are dimensionless, apart from the time $t$, in units of period $T$.Reuse & Permissions

Figure 4

Multifractal exponents (top) $D_1$ and (bottom) $D_2$ vs $t$, extracted from 1000 random phase realizations of size $N=2^{13}$ by the BC method with box sizes from 1 to 64. Red bottom curves: $\gamma =0.05$; blue top curves: $\gamma =0.5$; green middle curves: $\gamma =0.95$. The inset in the top panel shows time $\tau$ [defined by $D_1\left(\tau \right)=D_{1,\mathrm{as}}/2$] vs $\gamma$. Here $D_{1,\mathrm{as}}$ is the mean value of $D_1\left(t\right)$ in the time interval $[3000,5000]$. The solid curve is the best fit of the form $\exp [A/(B-\gamma )],$ with $A=0.749$ and $B=1.103$. The inset in the bottom panel is the mean second moment vs time for $\gamma =0.95$ (green squares), $0.05$ (orange crosses), $0.99$ (magenta circles), and $0.01$ (blue triangles). Black dashed lines are the analytical formula for the second moment (see text). All quantities on the figure are dimensionless, apart from the time $t$, in units of period $T$.Reuse & Permissions