热力学区域下随机单纯复形的谱研究
On the spectrum of Random Simplicial Complexes in Thermodynamic Regime
AI总结：
本文研究热力学区域下Linial-Meshulam复形邻接矩阵的谱，证明极限谱分布存在且带符号与不带符号情形互为镜像，给出其性质与大λ下的半圆律收敛性，并推导线图局部弱极限及其对谱连续部分的影响。
摘要：
Linial-Meshulam复形是定义在n个顶点上的随机单纯复形，具有完备的(d-1)维骨架，且d维单形以概率p独立出现，它是Erdos-Renyi随机图在高维下被研究最多的推广形式之一。\n本文讨论了当np→λ时Linial-Meshulam复形邻接矩阵的谱性质。我们证明了非随机极限谱分布（LSD）的存在性，并表明Linial-Meshulam复形的带符号与不带符号邻接矩阵的LSD互为镜像。我们还证明该LSD关于零不对称、无界，且在1/√(λd)的归一化下，当λ→∞时收敛于标准半圆律。论文后半部分推导了Linial-Meshulam复形线图的局部弱极限，并研究了其对LSD连续部分的影响。
英文摘要：
Linial-Meshulam complex is a random simplicial complex on $n$ vertices with a complete $(d-1)$-dimensional skeleton and $d$-simplices occurring independently with probability p. Linial-Meshulam complex is one of the most studied generalizations of the Erdos-Renyi random graph in higher dimensions. In this paper, we discuss the spectrum of adjacency matrices of the Linial-Meshulam complex when $np \rightarrow λ$. We prove the existence of a non-random limiting spectral distribution(LSD) and show that the LSD of signed and unsigned adjacency matrices of Linial-Meshulam complex are reflections of each other. We also show that the LSD is unsymmetric around zero, unbounded and under the normalization $1/\sqrt{λd}$, converges to standard semicircle law as $λ\rightarrow \infty$. In the later part of the paper, we derive the local weak limit of the line graph of the Linial-Meshulam complex and study its consequence on the continuous part of the LSD.
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