  We consider the branching random walk on the real line where the underlying
motion is of a simple random walk and branching is at least binary and at most
decaying exponentially in law. It is well known that the normalized empirical
measure converges to the Gaussian distribution for typical sets A. We therefore
analyze the probability that at step n the empirical distribution differs from
the Gaussian distribution by a constant \epsilon. We show that the decay is
doubly exponential in either n or \sqrt{n}, depending on the set A and
\epsilon, and we find the leading coefficient in the top exponent. To the best
of our knowledge, this is the first time such large deviation probabilities are
treated in this model.
