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

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 behaves as . 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 . 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.