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

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 $\mathbf{\zeta }$, there exists a critical offspring distribution $\tilde{\mathbf{\zeta }}$ such that the trees with offspring distribution $\mathbf{\zeta }$ and $\tilde{\mathbf{\zeta }}$ 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.