Asymptotics of the Longest Increasing Subsequence in Random Permutations

TL;DR

This paper explores the asymptotic behavior of the longest increasing subsequence in random permutations, highlighting classical results, limiting shapes, and fluctuation distributions, with an emphasis on conceptual understanding and visual explanations.

Contribution

It unifies classical proofs related to LIS asymptotics into a single narrative, providing intuitive explanations and visual aids, and reviews key theorems linking LIS to limiting distributions.

Findings

Expected LIS length grows as 2√n

Limiting shape of Young diagrams determined by variational principles

Fluctuations of LIS follow Tracy--Widom distribution

Abstract

In this paper, we examine the asymptotic behavior of the longest increasing subsequence (LIS) in a uniformly random permutation of $n$ elements. We rely on the Robinson--Schensted--Knuth correspondence, Young tableaux, and key classical results -- including the Erd\H{o}s--Szekeres theorem and the Hook Length Formula -- to demonstrate that the expected LIS length grows as $2\sqrt{n}$. We review the essential variational principles of Logan--Shepp and Vershik--Kerov, which determine the limiting shape of the associated random Young diagrams, and summarize the Baik--Deift--Johansson theorem that links fluctuations of the LIS length to the Tracy--Widom distribution. Our approach focuses on providing conceptual and intuitive explanations of these results, unifying classical proofs into a single narrative and supplying fresh visual examples, while referring the reader to the original literature for detailed proofs and rigorous arguments.