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

18 pages