  We consider the one-dimensional lattice model of interacting fermions with
disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111
(2007)]. To characterize a possible many-body localization transition as a
function of the disorder strength $W$, we use an exact renormalization
procedure in configuration space that generalizes the Aoki real-space RG
procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C13,
3369 (1980)]. We focus on the statistical properties of the renormalized
hopping $V_L$ between two configurations separated by a distance $L$ in
configuration space (distance being defined as the minimal number of elementary
moves to go from one configuration to the other). Our numerical results point
towards the existence of a many-body localization transition at a finite
disorder strength $W_c$. In the localized phase $W>W_c$, the typical
renormalized hopping $V_L^{typ} \equiv e^{\bar{\ln V_L}}$ decays exponentially
in $L$ as $ (\ln V_L^{typ}) \simeq - \frac{L}{\xi_{loc}}$ and the localization
length diverges as $\xi_{loc}(W) \sim (W-W_c)^{-\nu_{loc}}$ with a critical
exponent of order $\nu_{loc} \simeq 0.5$. In the delocalized phase $W<W_c$, the
renormalized hopping remains a finite random variable as $L \to \infty$, and
the typical asymptotic value $V_{\infty}^{typ} \equiv e^{\bar{\ln V_{\infty}}}$
presents an essential singularity $(\ln V_{\infty}^{typ}) \sim -
(W_c-W)^{-\kappa}$ with an exponent of order $\kappa \sim 1.4$. Finally, we
show that this analysis in configuration space is compatible with the
localization properties of the simplest two-point correlation function in real
space.
