Permuton limit of a generalization of the Mallows and $k$-card-minimum models

TL;DR

This paper introduces a new permutation model generalizing existing models, calculates its permuton limit, and proves universality results about its structure, including convergence to the logistic distribution under certain conditions.

Contribution

It defines a new generalized permutation model, computes its permuton limit, and confirms a conjecture about the universality of its band structure.

Findings

Permuton limit computed for the new model.

Law of large numbers for pattern densities established.

Universality of the band structure confirmed, with convergence to logistic distribution.

Abstract

We introduce and study a new random permutation model that generalizes the $k$-card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce the law of large numbers for pattern densities. Moreover, we prove a universality result about the band structure of the limiting permuton, confirming a conjecture of Travers about the $k$-card minimum model. More specifically, we show that if a certain model parameter goes to infinity then the appropriately scaled restriction of the permuton measure to a line that intersects the diagonal perpendicularly converges weakly to the logistic distribution.