  We consider branching random walks in $d$-dimensional integer lattice with
time-space i.i.d. offspring distributions. When $d \ge 3$ and the fluctuation
of the environment is well moderated by the random walk, we prove a central
limit theorem for the density of the population, together with upper bounds for
the density of the most populated site and the replica overlap. We also discuss
the phase transition of this model in connection with directed polymers in
random environment.
