I will be happy already with the simple euclidienne metric. And one can think V or W as $\mathbb{R}^n$, $\mathbb{S}^n$ or $\mathbb{T}^n$ if it does make the question simpler.

I guess this is the proof Kim talked about when he mention the binomial coefficients.

If $g$ has a very strong disorder, I think one should not be too surprise to see Anderson localisation phenomenom there. (But this is a completely behaviour.)

Isn't Euler Lagrange equation equivalent to Hamiltonian's and therefore implies conservation laws ? So I guess anything with a dissipative term, for example $\frac{d^2}{dt^2}q = -\frac{d}{dt}q$ should be a counter example.

I apologize for self advertising. In our paper arxiv.org/abs/2005.14180 we deal with this problem for the adjancy matrix of the E-R graph.

$f(x_1,x_2)=e^{i\pi x_1x_2}$ ?

Don't you mean : "Clearly if $A$ is locally $\epsilon-$dense then $A$ has positive measure"?

And you can have both: a permutation matrix with $\pm 1$ entries.

@ogogmad For physicists all matrices are diagonalizable except for the matrices than are not.

What about $D$ a permurtation matrix?

One condition I guess should be that $Y_m\neq Y_{m+1}$ a.s. otherwise there is no way one can see the time when $N_{t}=m+1$ occures.