AI 中文总结
该研究确定了本质上任意有限图上伯努利键渗流的临界概率为1/λ(G)，证实了相关猜想并推广了前人仅针对稠密图的定理，揭示了特定图上渗流的微妙行为。
AI 中文摘要
我们确定了本质上任意有限图上伯努利键渗流的临界概率。即，设λ(G)表示图G的谱半径（最大特征值），我们证明临界概率为1/λ(G)：高于该概率时，通常存在阶为Ω(λ(G))的连通分支；低于该概率时，所有连通分支的阶至多为O(√|G|)。这些结果特别证实了Krivelevich与Samotij关于给定平均度图上渗流的猜想，且极大推广了Bollobás、Borgs、Chayes和Riordan的定理——后者仅对稠密图证明了类似结果。我们的定理在多种 regime 下是最优的，还表明当谱半径约为最大度平方根时，渗流呈现出出人意料的微妙行为。
英文摘要
We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $λ(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/λ(G)$: above this probability there is typically a component of order $Ω(λ(G))$, whereas below it all components are of order at most $O(\sqrt{|G|})$. These results in particular confirm a conjecture of Krivelevich and Samotij about percolation on graphs of a given average degree, and vastly extend theorems of Bollobás, Borgs, Chayes, and Riordan, who proved analogous results but only for dense graphs. Our theorems are optimal in many regimes, and also demonstrate that percolation has an unexpectedly subtle behaviour on graphs whose spectral radius is roughly the square root of their maximum degree.
Comments22 pages
