Permutons from Demazure Products

TL;DR

This paper introduces new permuton families derived from Demazure products, analyzing their scaling limits and fluctuations, and connecting them to the KPZ universality class and tropical matrix multiplication.

Contribution

It constructs and characterizes new permutons from Demazure products, linking combinatorial, probabilistic, and tropical algebraic methods.

Findings

Demazure permutons from pipe dreams exhibit KPZ universality class behavior.

Explicit descriptions of limit permutons from bubble-sort operators.

Number of inversions in Demazure products of random permutations is asymptotically .

Abstract

We construct and analyze several new families of permutons arising from random processes involving the Demazure product on the symmetric group. First, we consider Demazure products associated to random pipe dreams, generalizing the Grothendieck permutons introduced by Morales, Panova, Petrov, and Yeliussizov by replacing staircase shapes with arbitrary order-convex shapes. Using the totally asymmetric simple exclusion process (TASEP) with geometric jumps, we prove precise scaling limit and fluctuation results for the associated height functions, showing that these models belong to the Kardar--Parisi--Zhang (KPZ) universality class. We then consider permutons obtained by applying deterministic sequences of bubble-sort operators to random initial permutations. We again provide precise descriptions of the limiting permutons. In a special case, we deduce the exact forms of the standard bubble-sort permutons, the supports of which were computed by DiFranco. A crucial tool in our analysis is a formulation, due to Chan and Pflueger, of the Demazure product as matrix multiplication in the min-plus tropical semiring. This allows us to define a Demazure product on the set of permutons. We discuss further applications of this product. For instance, we show that the number of inversions of the Demazure product of two independent uniformly random permutations of size $n$ is $\binom{n}{2}(1-o(1))$.