Scattering in Quantum Dots via Noncommutative Rational Functions

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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.

OriginalsprogEngelsk
TidsskriftAnnales Henri Poincare
Vol/bind22
Udgave nummer12
Sider (fra-til)4205-4269
Antal sider65
ISSN1424-0637
DOI
StatusUdgivet - 2021

Bibliografisk note

Funding Information:
L. Erdős and Yu. Nemish: Partially funded by ERC Advanced Grant RANMAT No. 338804. T. Krüger: Partially supported by VILLUM FONDEN research grant no. 29369 and the Hausdorff Center for Mathematics.

Publisher Copyright:
© 2021, The Author(s).

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