  We derive an effective quantum Josephson array model for a weakly interacting
one-dimensional condensate that is fragmented into weakly coupled puddles by a
disorder potential. The distribution of coupling constants, obtained from first
principles, indicate that weakly interacting bosons in a disorder potential
undergo a superfluid insulator transition controlled by a strong randomness
fixed point [Phys. Rev. Lett. 93, 150402 (2004)]. We compute renormalization
group flows for concrete realizations of the disorder potential to facilitate
finite size scaling of experimental results and allow comparison to the
behavior dictated by the strong randomness fixed point. The phase diagram of
the system is obtained with corrections to mean-field results.
