AI 中文总结
研究Linial-Meshulam随机复形上拉普拉斯算子特征值分布，定义高斯-半圆律并证明其极限分布几乎必然弱收敛于此律。
AI 中文摘要
我们研究了Linial-Meshulam模型$Y_{q+1}(n, p)$的标准化上拉普拉斯算子的经验特征值分布。首先，我们给出了高斯-半圆律的定义：$\mathcal{N}(0, \sigma^2)\boxplus \operatorname{SC}(s\sigma^2)$，并给出了其矩的配对划分组合公式。此外，我们还证明了该标准化上拉普拉斯算子的极限经验特征值分布遵循高斯-半圆律，即在几乎必然弱收敛的意义下。
英文摘要
We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model $Y_{q+1}(n, p)$. We first give a definition of a Gaussian--semicircle law: $\mathcal{N}(0, σ^2)\boxplus \operatorname{SC}(sσ^2)$, and give a combinatorial formula of its moments in terms of pairing partitions. In addition to that, we also prove that the limiting empirical eigenvalue distribution of this standardized up Laplacian follows a Gaussian--semicircle law, in the sense of almost surely weak convergence.
Comments24 pages, 7 figures
