Increasing subsequences of linear size in random permutations and the Robinson-Schensted tableaux of permutons

TL;DR

This paper extends the Robinson-Schensted correspondence to permutons, showing that permutations sampled from certain permutons have linearly growing longest increasing subsequences, with large deviation results and PDE characterizations.

Contribution

It introduces a permuton-based extension of the RS correspondence and demonstrates linear growth of LIS in this framework, along with large deviation and PDE results.

Findings

Permutons can induce linearly growing LIS in sampled permutations.

RS-tableaux of permutons satisfy a PDE.

Large deviation principles apply to LIS behavior.

Abstract

The study of longest increasing subsequences (LIS) in permutations led to that of Young diagrams via Robinson-Schensted's (RS) correspondence. In a celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams and found that the LIS of a uniform permutation of size n behaves as $2\sqrt{n}$. Independently and much later, Hoppen et al. introduced the theory of permutons as a scaling limit of permutations. In this paper, we extend in some sense the RS correspondence of permutations to the space of permutons. When the "RS-tableaux" of a permuton are non-trivial, we show that the RS-tableaux of random permutations sampled from this permuton exhibit a linear behavior, in the sense that their first rows and columns have lengths of linear order. In particular, the LIS of such permutations behaves as a multiple of n. We also prove some large deviation results for these convergences. Finally, by studying asymptotic properties of Fomin's algorithm for permutations, we show that the RS-tableaux of a permuton satisfy a partial differential equation.