Yaglom limit for critical neutron transport

We consider the classical Yaglom limit theorem for the Neutron Branching Process (NBP) in the setting that the mean semigroup is critical, i.e. its leading eigenvalue is zero. We show that the law of the process conditioned on survival is asymptotically equivalent to an exponential distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of spatial branching processes and superprocesses, as well as in the setting of isotropic homogeneous Neutron Branching Processes back in the 1970s, a large component of our proof appeals to a completely new combinatorial approach which is necessary as, unlike the aforementioned literature, the branching mechanism is non-local and anisotropic. Our new approach and the main novelty of this work is based around a coupled pair of inductive hypotheses, which describe the convergence of all of the associated moments of the NBP. For the proof of the latter, we make use of the recently demonstrated spine decomposition, cf. \cite{SNTE}, pulling out the leading order terms using a non-trivial combinatorial decomposition.