Simultaneous large deviations for the shape of Young diagrams associated with random words

TL;DR

This paper studies the large deviations in the shape of Young diagrams derived from random words, revealing connections to random matrix spectra and providing concentration bounds for the longest increasing subsequence.

Contribution

It establishes large deviation principles for Young diagram shapes associated with random words, linking them to spectral distributions of GUE and providing non-asymptotic bounds.

Findings

Large deviations match the spectrum of traceless GUE for uniform letter distribution.

Control of probabilities ensures large deviation principles in non-uniform cases.

Non-asymptotic concentration bounds for the longest increasing subsequence.

Abstract

We investigate the large deviations of the shape of the random RSK Young diagrams associated with a random word of size $n$ whose letters are independently drawn from an alphabet of size $m=m(n)$. When the letters are drawn uniformly and when both $n$ and $m$ converge together to infinity, $m$ not growing too fast with respect to $n$, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. In the non-uniform case, a control of both highest probabilities will ensure that the length of the top row of the diagram satisfies a large deviation principle. In either case, both speeds and rate functions are identified. To complete our study, non-asymptotic concentration bounds for the length of the top row of the diagrams, that is, for the length of the longest increasing subsequence of the random word are also given for both models.