Asymptotics for the Length of the Longest Increasing Subsequence of Binary Markov Random Word

TL;DR

This paper proves that the scaled distribution of the longest increasing subsequence length in a binary Markov chain converges to the distribution of the maximal eigenvalue of a 2x2 Gaussian matrix, revealing a spectral limit shape.

Contribution

It provides a new elementary proof connecting the longest increasing subsequence in a binary Markov chain to Gaussian matrix eigenvalues, with a focus on asymptotic behavior.

Findings

Limiting law of LI_n is the maximal eigenvalue of a 2x2 Gaussian matrix.

The shape of the associated Young diagrams converges to the spectrum of this matrix.

Elementary combinatorial and invariance principles underpin the proof.

Abstract

Let $(X_n)_{n\ge 0}$ be an irreducible, aperiodic, and homogeneous binary Markov chain and let $LI_n$ be the length of the longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$. Using combinatorial constructions and weak invariance principles, we present elementary arguments leading to a new proof that (after proper centering and scaling) the limiting law of $LI_n$ is the maximal eigenvalue of a $2 \times 2$ Gaussian random matrix. In fact, the limiting shape of the RSK Young diagrams associated with the binary Markov random word is the spectrum of this random matrix.