  The generalization of the Dyson Brownian Motion approach of random matrices
to Anderson Localization (AL) models [Chalker, Lerner and Smith PRL 77, 554
(1996)] and to Many-Body Localization (MBL) Hamiltonians [Serbyn and Moore
arxiv:1508.07293] is revisited to extract the level repulsion exponent $\beta$,
where $\beta=1$ in the delocalized phase governed by the Wigner-Dyson
statistics, $\beta=0$ in the localized phase governed by the Poisson
statistics, and $0<\beta_c<1$ at the critical point. The idea is that the
Gaussian disorder variables $h_i$ are promoted to Gaussian stationary processes
$h_i(t)$ in order to sample the disorder stationary distribution with some time
correlation $\tau$. The statistics of energy levels can be then studied via
Langevin and Fokker-Planck equations. For the MBL quantum spin Hamiltonian with
random fields $h_i$, we obtain $\beta =2q^{EA}_{n,n+1}(N)/q^{EA}_{n,n}(N) $ in
terms of the Edwards-Anderson matrix $q^{EA}_{nm}(N) \equiv \frac{1}{N}
\sum_{i=1}^N | < \phi_n | \sigma_i^z | \phi_m> |^2 $ for the same eigenstate
$m=n$ and for consecutive eigenstates $m=n+1$. For the Anderson Localization
tight-binding Hamiltonian with random on-site energies $h_i$, we find $\beta =2
Y_{n,n+1}(N)/(Y_{n,n}(N)-Y_{n,n+1}(N)) $ in terms of the Density Correlation
matrix $Y_{nm}(N) \equiv \sum_{i=1}^N | < \phi_n | i> |^2 | <i | \phi_m> |^2 $
for consecutive eigenstates $m=n+1$, while the diagonal element $m=n$
corresponds to the Inverse Participation Ratio $Y_{nn}(N) \equiv \sum_{i=1}^N |
< \phi_n | i> |^4 $ of the eigenstate $| \phi_n>$.
