Erdos L, Krueger T, Nemish Y (2021)

**Publication Type:** Journal article

**Publication year:** 2021

**DOI:** 10.1007/s00023-021-01085-6

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.

IST Austria
Austria (AT)
University of Copenhagen
Denmark (DK)
University of California, San Diego
United States (USA) (US)

**APA:**

Erdos, L., Krueger, T., & Nemish, Y. (2021). Scattering in Quantum Dots via Noncommutative Rational Functions. *Annales Henri Poincaré*. https://doi.org/10.1007/s00023-021-01085-6

**MLA:**

Erdos, Laszlo, Torben Krueger, and Yuriy Nemish. "Scattering in Quantum Dots via Noncommutative Rational Functions." *Annales Henri Poincaré* (2021).

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