论文标题
变形的单个环定理
Deformed single ring theorems
论文作者
论文摘要
给定一系列确定性矩阵$ a = a_n $和一系列确定性的非负矩阵$σ=σ_n$，使得$ a \ to $ a \ to $ a \ to a $ a \ in $ \ ast $ \ ast $ - distribution in Boy-ast $ - distribution in Boy a $ a $ a $ a $ and $ neumann neumann algebra $ n n neumann algebra $ \ arvac}令$ u = u_n $和$ v = v_n $是独立的haar分布统一矩阵。 We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $UΣV^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $σ$.如果$ a $是Hermitian或单一的，则可以删除假设。通过将$ a = 0 $放置，我们的结果消除了Guionnet，Krishnapur和Zeitouni的单个环定理中的规律性假设。我们还证明了最佳尺度上的局部收敛性，从而扩展了BAO，ERDőS和Schnelli的局部单个环定理。
Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $Σ=Σ_N$ such that $A\to a$ and $Σ\to σ$ in $\ast$-distribution for some operators $a$ and $σ$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $UΣV^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $σ$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli.
