随机正则图的体相普适性
Bulk universality of random regular graphs
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中文总结 AI 辅助
本文证明均匀随机d-正则图邻接矩阵的体相特征值点过程收敛于Sine_1过程，并建立间隙与相关测度的普适性，方法基于四阶矩估计与切换恒等式。
中文摘要 AI 辅助
我们考虑在N个顶点上的均匀随机简单d-正则图的邻接矩阵。对于每个固定的d≥3和每个固定的体相能量，我们证明重标度后的特征值点过程收敛到强度为1/π的Sine_1过程。我们还建立了在确定性体相标签处连续间隙的普适性以及固定能量相关测度的普适性。特征向量输入包括在均匀采样星形图上混合四阶矩的多项式估计。这些估计防止了Backhausz--Szegedy定性联合高斯极限中方差的损失，从而为附近能量处的固定元组产生独立的方差为一的高斯波。精确的切换恒等式随后给出平稳标记流和微观环层次结构，这确定了点过程极限。附录使用计数矩界和标记极限的条件对数气体定律证明了更强的相关性和间隙陈述。
英文摘要
We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/π$. We also establish universality of consecutive gaps at deterministic bulk labels and of fixed-energy correlation measures. The eigenvector input consists of polynomial estimates for mixed fourth moments on a uniformly sampled star. These estimates prevent loss of variance in the qualitative joint Gaussian limits of Backhausz--Szegedy, yielding independent variance-one Gaussian waves for fixed tuples at nearby energies. Exact switching identities then give a stationary marked flow and the microscopic loop hierarchy, which identifies the point-process limit. The appendices prove the stronger correlation and gap statements using counting-moment bounds and conditional log-gas laws for tagged limits.
发表机构
- Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
- Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
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