## 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.

- Speaker: Nina Gantert (Technical University of Munich)
- Tuesday 12 March 2019, 14:00–15:00
- Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
- Series: Probability; organiser: Perla Sousi.