Large deviations for the maximum of a branching random walk

With Nina Gantert (Technical University of Munich)

Large deviations for the maximum of a branching random walk

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.
As time permits, we also give some results about branching random walks
in random environment.

The talks is based on joint work with Thomas Höfelsauer.

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