Exercise given at the end of Leure.
Letµbe a critical offspring diribution onZ+, with finite and positive varianceσ. Recall that for every treeτandk≥, [τ]k ={u∈τ;|u| ≤k}is the treeτ cut above generationk, and thatT∞ denotes theµ-Galton–Watson tree conditioned to survived.
. Let Tn be a Galton–Watson tree with offspring diribution µ, conditioned on having n leaves. Show that Tn converges locally in diribution to T∞ as n → ∞, that is for every finite treeτandk≥,
P([Tn]k =τ) −→
n→∞ P([T∞]k =τ)
. Let Dn be a random disseion of {, eiπ/n, . . . , ei(n−)π/n}, chosen uniformly at random among all disseions of{, eiπ/n, . . . , ei(n−)π/n}.
(i) LetFndenote the degree of the face adjacent to the side [, eiπ/n] ofDn. Show that
P(Fn=k) −→
n→∞ (k−) −√
!k−
, k≥.
(ii) Let Vn denote the number of diagonals adjacent to the vertex corresponding to the complex numberinDn. Show that
P(Vn=k) −→
n→∞ (k+)·(−
√)(
√−)k, k≥.