@article{9912,
  abstract     = {In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁≪𝑀 channels, the density 𝜌 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 𝜙:=𝑁/𝑀≤1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 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 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−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.},
  author       = {Erdös, László and Krüger, Torben H and Nemish, Yuriy},
  issn         = {1424-0661},
  journal      = {Annales Henri Poincaré },
  pages        = {4205–4269},
  publisher    = {Springer Nature},
  title        = {{Scattering in quantum dots via noncommutative rational functions}},
  doi          = {10.1007/s00023-021-01085-6},
  volume       = {22},
  year         = {2021},
}

@article{7512,
  abstract     = {We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.},
  author       = {Erdös, László and Krüger, Torben H and Nemish, Yuriy},
  issn         = {1096-0783},
  journal      = {Journal of Functional Analysis},
  number       = {12},
  publisher    = {Elsevier},
  title        = {{Local laws for polynomials of Wigner matrices}},
  doi          = {10.1016/j.jfa.2020.108507},
  volume       = {278},
  year         = {2020},
}

@article{6240,
  abstract     = {For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.},
  author       = {Alt, Johannes and Erdös, László and Krüger, Torben H and Nemish, Yuriy},
  issn         = {0246-0203},
  journal      = {Annales de l'institut Henri Poincare},
  number       = {2},
  pages        = {661--696},
  publisher    = {Institut Henri Poincaré},
  title        = {{Location of the spectrum of Kronecker random matrices}},
  doi          = {10.1214/18-AIHP894},
  volume       = {55},
  year         = {2019},
}

@article{1023,
  abstract     = {We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.},
  author       = {Nemish, Yuriy},
  issn         = {1083-6489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Local law for the product of independent non-Hermitian random matrices with independent entries}},
  doi          = {10.1214/17-EJP38},
  volume       = {22},
  year         = {2017},
}