Show that Tn converges locally in diribution to T∞ as n

(1)

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 T_n be a Galton–Watson tree with offspring diribution µ, conditioned on having n leaves. Show that T_n converges locally in diribution to T_∞ as n → ∞, that is for every finite treeτandk≥,

P([T_n]_k =τ) −→

n→∞ P([T_∞]_k =τ)

. Let D_n be a random disseion of {, e^iπ/n, . . . , e^i(n⁻^)π/n}, chosen uniformly at random among all disseions of{, e^iπ/n, . . . , e^i(n⁻^)π/n}.

(i) LetF_ndenote the degree of the face adjacent to the side [, e^iπ/n] ofD_n. Show that

P(F_n=k) −→

n→∞ (k−) −√



!k−

, k≥.

(ii) Let V_n denote the number of diagonals adjacent to the vertex corresponding to the complex numberinD_n. Show that

P(V_n=k) −→

n→∞ (k+)·(−

√)^(

√−)^k, k≥.

