Asymptotic properties of expansive galton-watson trees

We consider a super-critical Galton-Watson tree τ whose non-degenerate offspring distribution has finite mean. We consider the random trees τ_n distributed as τ conditioned on the n-th generation, Z_n, to be of size a_n ∈ N. We identify the possible local limits of τ_n as n goes to infinity according to the growth rate of a_n. In the low regime, the local limit τ⁰ is the Kesten tree, in the moderate regime the family of local limits, τ^θ for θ ∈ (0,+∞), is distributed as τ conditionally on {W=θ}, where W is the (non-trivial) limit of the renormalization of Z_n. In the high regime, we prove the local convergence towards τ^∞ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits (τ^θ, θ ∈ [0,∞]).