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?