稀释随机图的秩
The rank of diluted random graphs
摘要：
研究了大稀疏随机图的邻接矩阵的秩：对于局部收敛到Galton-Watson树$T$（GWT）的图序列$(G_n)_{n\geq0}$，我们给出了特征值0的渐近重数关于$T$的度母函数$ϕ_*$的显式表达式.在第一部分中，我们证明了与$T$相关的邻接算子总是自伴的;我们分析了根处相关的谱测度，并刻画了其原子质量在0.在第二部分中，我们建立了一个关于$ϕ_*$的充分条件，使得这个原子质量的期望恰好是$(G_n)_{n\geq 0}$邻接矩阵的核的维数的归一化极限。我们的证明借鉴了算法分析、泛函分析、随机矩阵理论和统计物理的思想。
英文摘要：
We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs $(G_n)_{n\geq0}$ converging locally to a Galton--Watson tree $T$ (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function $ϕ_*$ of $T$. In the first part, we show that the adjacency operator associated with $T$ is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on $ϕ_*$ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of $(G_n)_{n\geq 0}$. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
