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?