Scattering in quantum dots via noncommutative rational functions

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 $\phi \to 0$ 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.