Poisson limit theorems for the Robinson-Schensted correspondence and for the multi-line Hammersley process

TL;DR

This paper proves multidimensional Poisson limit theorems for the Robinson-Schensted-Knuth algorithm applied to random inputs, extending previous results to multiple rows and connecting to the stationary distribution of the multi-line Hammersley process.

Contribution

It extends the Poisson limit theorem for the Robinson-Schensted-Knuth algorithm to multiple rows, linking it to the stationary distribution of the multi-line Hammersley process.

Findings

Multidimensional Poisson limit theorem established.

Convergence of the multi-line Hammersley process to its stationary distribution.

Extension of Aldous and Diaconis's results to multiple rows.

Abstract

We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.