Critical exponential tiltings for size-conditioned multitype Bienaym\'e--Galton--Watson trees

We consider here multitype Bienaym\'e--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 Bienaym\'e--Galton--Watson trees.Comment: 18 page