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 β 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, β non-negative integers and finite matrix size. For β = 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.

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Exact eigenvalue order statistics for the reduced density matrix of a bipartite system

“…We would like to thank Arul Lakshminarayan for drawing our attention to Ref. [18]. This work was supported in part by a seed grant from IIT Madras to the Centre for Quantum Information Theory of Matter and Spacetime, under the IoE-CoE scheme.…”

Joint moments of a characteristic polynomial and its derivative for the circular $β$-ensemble

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