Critical exponential tiltings for size-conditioned multitype Bienaymé--Galton--Watson trees

Abstract

We consider here multitype Bienaymé--Galton--Watson trees, under the conditioning that the numbers of vertices of given type satisfy some linear relations. We prove that, under some smoothness conditions on the offspring distribution $\mathbfζ$, there exists a critical offspring distribution $\tilde{\mathbfζ}$ such that the trees with offspring distribution $\mathbfζ$ and $\tilde{\mathbfζ}$ have the same law under our conditioning. This allows us in a second time to characterize the local limit of such trees, as their size goes to infinity. Our main tool is a notion of exponential tilting for multitype Bienaymé--Galton--Watson trees.