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

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