Permutons from Demazure Products

TL;DR

The paper investigates permutons arising from Demazure products in the symmetric group, connecting random pipe dreams and TASEP-based scaling limits to deterministic bubble-sort dynamics. It introduces a min-plus tropical formulation of the Demazure product on permutons and proves convergence and KPZ-fluctuation results for several model families, including peridot and Polyphemus permutons arising from rectangle and trapezoid shapes. It also develops a comprehensive framework for Demazure products of permutons, enabling explicit descriptions of limiting objects and a rich zoo of examples, along with a memory- and structure-driven interpretation in the rectangle/bubble-sort settings. The work highlights deep connections between combinatorial models, interacting particle systems, and scaling limits, offering both exact limits and broad methodological tools for constructing and analyzing new permuton universes with potential applications in probability and geometry. Overall, the paper extends the theory of permutons via Demazure products, yielding precise limit shapes, fluctuation results in the KPZ class, and a versatile toolkit for generating and analyzing a wide array of limiting permutons.

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 $\binom{n}{2}(1-o(1))$.

Figures (15)

• Figure 1: On the left is a pipe dream in an order-convex shape $\mathscr{S}(\Lambda_{\swarrow},\Lambda_{\nearrow})$, where the lattice paths $\Lambda_{\swarrow},\Lambda_{\nearrow}\in\boldsymbol{\Lambda}_9$ are drawn in red and green. The image on the right computes the Demazure product of the corresponding subword by resolving the three crossings in the shaded boxes. This Demazure product is $134265897$.
• Figure 2: A random pipe dream with parameter $p=1/2$ in a $20\times 40$ rectangle shape $\mathscr{S}$ with crossings resolved in the shaded boxes. Different pipes receive different colors. The Demazure product $\Delta_p(\mathscr{S})$ is obtained by reading the numbers along the northeast boundary.
• Figure 3: On the left is the plot of the random permutation $\Delta_{1/2}(\mathscr{S})\in\mathfrak{S}_{60}$ from \ref{['fig:rectangle_example']}. In the middle and on the right are plots of the random permutations $\Delta_{1/2}(\mathscr{S})$ and $\Delta_{3/4}(\mathscr{S})$ in $\mathfrak{S}_{2400}$, where $\mathscr{S}$ is an $800\times 1600$ rectangle shape.
• Figure 4: The plots of $\tau_{(1,2,\ldots,1999)}^{400k}(u)$ for $k=0,1,2,3,4,5$, where $u$ is a uniformly random permutation in $\mathfrak{S}_{2000}$.
• Figure 5: On the left is a shape representing the commutation class of the Coxeter word ${\bf c}_9=(1,3,2,7,6,5,4,8)$ for $\mathfrak{S}_9$. This shape is uniquely determined by the lattice path $\Lambda_{\swarrow}^{(9)}$. On the right is the shape $\mathscr{S}^{(9)}$ such that $\tau_{{\bf w}(\mathscr{S}^{(9)})}=\tau_{{\bf c}_9}^4$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Proposition 2.1: HKMRS
• Proposition 2.2: HKMRS