[Paper Review] A note on GUE minors, maximal Brownian functionals and longest increasing subsequences

Florent Benaych-Georges, Christian Houdré|arXiv (Cornell University)|Dec 11, 2013

Random Matrices and Applications8 references1 citations

TL;DR

This paper establishes stochastic equalities between the spectra of GUE matrix minors and maximal functionals of independent Brownian motions, enabling the derivation of the limiting distribution of the longest increasing subsequence in a random word via the RSK correspondence and the limiting shape of associated Young diagrams.

ABSTRACT

We present equalities in law between the spectra of the 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.

Motivation & Objective

• To establish equalities in law between spectra of GUE matrix minors and maximal functionals of independent Brownian motions.
• To connect these stochastic equalities to the asymptotic behavior of RSK Young diagrams associated with random words.
• To derive the limiting distribution of the length of the longest increasing subsequence in a random word using spectral and stochastic methods.
• To provide a new probabilistic derivation of the Tracy-Widom distribution for the longest increasing subsequence via GUE minors.

Proposed method

• Uses the joint distribution of eigenvalues of GUE matrix minors to relate them to functionals of Brownian motion.
• Applies the RSK correspondence to map random words to Young diagrams, linking combinatorics to random matrix theory.
• Employs stochastic calculus and scaling limits to connect the maximal functionals of Brownian motions to the top row of Young diagrams.
• Applies known results on the limiting shape of Young diagrams under proper centering and scaling to derive asymptotic laws.
• Uses the fact that the length of the top row of the Young diagram equals the length of the longest increasing subsequence.
• Relies on the equivalence in law between GUE minor spectra and Brownian functional maxima to transfer limiting behavior.

Experimental results

Research questions

• RQ1 How are the spectra of GUE matrix minors related to maximal functionals of independent Brownian motions?
• RQ2 What is the limiting shape of RSK Young diagrams associated with a random word, under proper centering and scaling?
• RQ3 How does the length of the longest increasing subsequence in a random word behave asymptotically?
• RQ4 Can the limiting distribution of the longest increasing subsequence be derived from GUE minor spectra?
• RQ5 What is the role of the RSK correspondence in connecting random matrix theory to combinatorics of increasing subsequences?

Key findings

• The spectra of GUE matrix minors are equal in distribution to certain maximal functionals of independent Brownian motions.
• The limiting shape of the RSK Young diagrams for a random word is derived from the spectral properties of GUE minors.
• The length of the longest increasing subsequence in a random word converges to the Tracy-Widom distribution in the limit, as a consequence of the spectral-stochastic equivalence.
• The top row of the Young diagram, which encodes the longest increasing subsequence, inherits its limiting law from the maximal functional of Brownian motion.
• The connection between GUE minors and Brownian functionals provides a new probabilistic route to the asymptotic distribution of longest increasing subsequences.
• The results unify aspects of random matrix theory, Brownian motion, and combinatorics via the RSK correspondence.