Kalle Eriksson: Interface fluctuations in a two-type annihilating branching random walk with step initial configuration

Abstract: Two types of particles start separated on the integer line: one blue particle is placed at each negative site and one red particle at each nonnegative site. Both types of particles evolve according to a branching random walk in continuous time, and particles of opposite colors annihilate whenever they occupy the same site. This competition raises a natural question: how does the interface between the two populations fluctuate over time?

We show that the relevant fluctuation scale is \(t^{1/4}\). We first establish that, even when the branching dynamics do not preserve ordering (blue to the left of red), the configuration of particles is ordered with probability tending to one as time tends to infinity. This allows us to interpret the midpoint between the rightmost blue particle and the leftmost red particle as an interface. We then show that this interface makes excursions on the scale \(t^{1/4}\) away from a deterministic drift line, with probabilities bounded away from zero for sufficiently large times. We also provide an upper bound on the scale of these fluctuations. Finally, these fluctuations occur in both directions infinitely often, almost surely.

This is joint work with my supervisor Daniel Ahlberg.