Burning Random Trees

TL;DR

This paper analyzes the burning number $b(T_n)$ of conditioned Galton–Watson trees $T_n$ with offspring distribution $\xi$ satisfying $E[\xi]=1$ and $0<\mathrm{Var}[\xi]=\sigma^{2}<\infty$. By reducing burning to a ball-covering problem and employing both combinatorial bounds and probabilistic analysis of conditioned trees via DFS/Lukasiewicz representations, the authors prove that $b(T_n)$ scales as $n^{1/3}$ in probability: specifically, for any $\varepsilon(n)\to0$, $(\varepsilon n)^{1/3} \le b(T_n) \le (n/\varepsilon)^{1/3}$ with probability $1-O(\varepsilon)$. The lower bound relies on a bound for the expected number of vertex pairs at fixed distances, while the upper bound uses a deterministic covering strategy with balls of radius $2k$ and a probabilistic height bound for subtrees. The results place the burning process on random simply generated trees in a precise $n^{1/3}$-scaling regime and suggest avenues toward a limit theorem and universal constants involving $\sigma^{2}$.

Abstract

Let $\mathcal{T}$ be a Galton-Watson tree with a given offspring distribution $ξ$, where $ξ$ is a $Z_{\geq 0}$-valued random variable with $E[ξ] = 1$ and $0 < σ^{2}:=Var[ξ] < \infty$. For $n \geq 1$, 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 $n^{1/3}$.

Figures (2)

• Figure 1: A tree $T$ (a) and a cycle corresponding to a depth-first search of $T$ (b). Any burning sequence for the cycle projects to a burning sequence for the tree.
• Figure 2: A rooted tree $\tau$ with preorder degree sequence $(1,2,2,0,0,1,0)$ (a) and its lattice path representation (b).

Theorems & Definitions (15)

• Conjecture 1.1
• Theorem 1.2
• Theorem 3.1: devroye2011distances
• Proposition 3.2
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• Lemma 4.3: janson2012simply
• Lemma 4.4: janson2012simply