  We consider real-valued branching random walks and prove a large deviation
result for the position of the rightmost particle. The position of the
rightmost particle is the maximum of a collection of a random number of
dependent random walks. We characterise the rate function as the solution of a
variational problem. We consider the same random number of independent random
walks, and show that the maximum of the branching random walk is dominated by
the maximum of the independent random walks. For the maximum of independent
random walks, we derive a large deviation principle as well. It turns out that
the rate functions for upper large deviations coincide, but in general the rate
functions for lower large deviations do not.
