
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…
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Taxonomy
TopicsAdvanced Algebra and Logic · Fuzzy and Soft Set Theory · semigroups and automata theory
