论文标题
光谱差距和边缘普遍存在
Spectral gap and edge universality of dense random regular graphs
论文作者
论文摘要
令$ \ mathcal a $为$ n $ vertices上随机$ d $ regular Graph的邻接矩阵，我们用$λ_1\ geqλ_2\ cdots \geqλ__{n} $表示其特征值。对于$ n^{2/3} \ ll d \ leq n/2 $，我们证明了$ \ mathcal a $的极端特征值的最佳刚度估计，这尤其暗示\ [\ max \ [| max \ {|λ_n|，λ_2\} <2 \} <2 \ sqrt probibity在$ d $的同一方案中，我们还表明，\ [n^{2/3} \ bigG（\ frac {λ_2+d/n} $ \ mathrm {tw} _1 $是Goe的tracy-widom发行版；类似物的结果也适用于其他非平凡的极端特征值。
Let $\mathcal A$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $λ_1\geq λ_2\cdots \geq λ_{N}$. For $N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of the extreme eigenvalues of $\mathcal A$, which in particular imply that \[ \max\{|λ_N|,λ_2\} <2\sqrt{d-1} \] with overwhelming probability. In the same regime of $d$, we also show that \[ N^{2/3}\bigg(\frac{λ_2+d/N}{\sqrt{d(N-d)/N}}-2\bigg) \overset{d}{\longrightarrow} \mathrm{TW}_1\,, \]where $\mathrm{TW}_1$ is the Tracy-Widom distribution for GOE; analogues results also hold for other non-trivial extreme eigenvalues.
