Critical branching processes in random environment with immigration and an application to randomised reproducing graphs

We study branching processes in an i.i.d.\ random environment with immigration in the critical regime, where the underlying offspring mechanism satisfies the critical condition that the log of the average population growth, across environments, and before immigration, is zero. In this setting environmental fluctuations are balanced on average, and the long-term behaviour is determined by the interaction between these fluctuations and the immigration sequence. While recurrence and transience criteria for critical BPREI were established by Bauernschubert (2014), the possibility of null recurrence remained unresolved.
We show that, under natural integrability assumptions on the offspring and immigration distributions, a critical BPREI is null recurrent. In particular, the process returns to zero infinitely often but admits no stationary distribution. Our results close a gap in the classification of the critical regime and provide a structural understanding of the balance between environmental variability and immigration.
As an application, we resolve the open critical case of the Randomised Reproducing Graph (`RRG') model introduced by Jordan (2011), showing that in the critical regime the proportion of vertices of a fixed degree admits no limiting distribution.