In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via \(N\ll M\) channels, the density \(\rho \) of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio \(\phi := N/M\le 1\); and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit \(\phi \rightarrow 0\), we recover the formula for the density \(\rho \) that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any \(\phi <1\) but in the borderline case \(\phi =1\) an anomalous \(\lambda ^{-2/3}\) singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.

在具有M 个内部自由度的量子点中传输的惯用随机矩阵模型中，通过\(N\ll M\)通道耦合到混沌环境，传输特征值的密度\(\rho \)由特定的不变量计算所有特征值的联合概率密度的显式公式可用的集成。我们在允许 (i) 任意比率\(\phi := N/M\le 1\)的大N 范围中重新审视这个问题；(ii) 量子点哈密顿量矩阵元素的一般分布。在极限\(\phi \rightarrow 0\) 中，我们恢复了密度\(\rho \)的公式Benakker (Rev Mod Phys 69:731–808, 1997) 推导出了一个特殊的矩阵集合。我们还证明，Beenakker 公式中密度在零和全透射时的平方根倒数奇点对于任何\(\phi <1\)仍然存在，但在边界情况\(\phi =1\)一个异常\(\lambda ^{-2/3}\)奇点出现在零处。为了获得这种级别的一般性，我们开发了关于具有 iid 项的大型随机矩阵中的一大类非交换有理表达式的谱密​​度的全局和局部定律理论。