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

## TL;DR

This paper introduces a recursion scheme for computing the largest eigenvalue distribution of the Wishart-Laguerre ensemble, with applications in quantum transport and communication, providing exact results where previous methods were limited.

## Contribution

It presents a novel recursion method for explicit eigenvalue distribution calculation, applicable to both original and fixed-trace models, improving over symbolic determinant-based approaches.

## Key findings

- Effective recursion scheme for eigenvalue distribution
- Exact Landauer conductance distribution solution
- Applications in quantum communication and entanglement

## 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.