Problem ------------------ Consider two *d* x *d* complex matrices, *R* and *S*, whose entries lie in the unit disk: $\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $. Say that *R* is constructed by randomly choosing complex numbers from the unit disk, but *S* is constructed as $\quad S_{i,j} = f(i/d,j/d)$ where $f(x,y)$ is a smooth function for $x,y \in [0,1]$, with $|f(x,y)|<1$. In other words, the entries of *S* are smooth functions of the indices (in the limit of large *d*), but those of *R* are not. Question ------------------ How do the trace norms $\quad ||R||=Tr[\sqrt{R^\dagger R}] \quad$ and $\quad ||S||=Tr[\sqrt{S^\dagger S}]$ of these matrices behave as $d \to \infty$ ? Numerical Evidence ------------------ A few lines of Mathematica strongly suggest that $\quad ||R|| \propto d^{3/2}$ but $\quad ||S|| \propto d$ for large *d*. (The proportionality constants depend on the probability distribution used to pick numbers from the unit disk for *R* and the function $f(x,y)$ used to pick entries for *S*, respectively.) What explains this behavior?