Recursion scheme for the largest $\beta$-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport

Abstract

The largest eigenvalue distribution of the Wishart-Laguerre ensemble, indexed by Dyson parameter $\beta$ and Laguerre parameter $a$, is fundamental in multivariate statistics and finds applications in diverse areas. Based on a generalization of the Selberg integral, we provide an effective recursion scheme to compute this distribution explicitly in both the original model, and a fixed-trace variant, for $a,\beta$ non-negative integers and finite matrix size. For $\beta =2$ this circumvents known symbolic evaluation based on determinants which become impractical for large dimensions. Our exact results have immediate applications in the areas of multiple channel communication and bipartite entanglement. Moreover, we are also led to the exact solution of a long standing problem of finding a general result for Landauer conductance distribution in a chaotic mesoscopic cavity with two ideal leads. Thus far, exact closed-form results for this were available only in the Fourier-Laplace space or could be obtained on a case-by-case basis.

Published version; 20 pages, 2 figures in the main text + 2 in the Mathematica code towards the end

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Cited by in corpus (10)

• Turbulence Hierarchy and Multifractality in the Integer Quantum Hall Transition
• Differential identities for the structure function of some random matrix ensembles
• Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Spectral statistics for the difference of two Wishart matrices
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
• Optical MIMO communication with unequal power allocation to channels
• Exact eigenvalue order statistics for the reduced density matrix of a bipartite system