  The extreme value statistics of active matter offer significant insight into
their unique properties. A phase transition has recently been reported in a
model of branching run-and-tumble particles, describing the spatial spreading
of an evolving colony of active matter in one-dimension. In a "persistent"
phase, the particles form macroscopic robust clusters that ballistically
propagate as a whole while in an "intermittent" phase, particles are isolated
instead. We focus our study on the fluctuations of the rightmost position
$x_{\max}(t)$ reached by time $t$ for this model. At long time, as the colony
progressively invades the unexplored region, the cumulative probability of
$x_{\max}(t)$ is described by a travelling front. The transition has a
remarkable impact on this front. In the intermittent phase it is qualitatively
similar to the front satisfying the Fisher-KPP equation, which famously
describes the extreme value statistics of the non-active branching Brownian
motion. A dramatically different behaviour appears in the persistent phase,
where activity imparts the front with unexpected and unusual features which we
compute exactly.
