Burning Random Trees

Let $\mathcal{T}$ be a Galton-Watson tree with a given offspring distribution $\xi$, where $\xi$ is a $Z_{\ge 0}$-valued random variable with $E[\xi ]=1$ and $0<{\sigma }^2:=Var[\xi ]<\mathrm{\infty }$. For $n\ge 1$, let $T_n$ be the tree $\mathcal{T}$ conditioned to have $n$ vertices. In this paper we investigate $b(T_n)$, the burning number of $T_n$. Our main result shows that asymptotically almost surely $b(T_n)$ is of the order of $n^{1\mathrm{/}3}$.