GUE Minors, Maximal Brownian Functionals and Longest Increasing Subsequences in Random Words

F. Benaych-Georges, C. Houdre'

2015, v.21, Issue 1, 109-126


We present equalities in law between the spectra of the principal
minors of a GUE matrix
and some maximal functionals of independent Brownian motions. In turn, these results
allow to recover the limiting shape (properly centered and scaled) of the RSK Young diagrams
associated with a random word as a function of the spectra of these minors.
Since the length of the top row of the diagrams is the length of the longest increasing
subsequence of the random word, the corresponding limiting law also follows.

Keywords: random matrices, Brownian motion, random words, longest increasing subsequences, RSK correspondence


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