Permutations with fixed pattern densities

TL;DR

This paper investigates the limiting behavior of random permutations constrained by fixed pattern densities, deriving their limit shapes through entropy maximization and exploring phase transitions.

Contribution

It introduces a method to determine permuton limit shapes with fixed pattern densities using entropy maximization, including numerical computations and phase transition analysis.

Findings

Derived explicit limit shapes for various fixed pattern densities.

Identified a phase transition in the case of fixed 123 and 321 densities.

Provided a dynamic construction to describe permutons.

Abstract

We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed \hbox{12} density, with fixed \hbox{12} and \hbox{123} densities, with fixed \hbox{12} density and the sum of \hbox{123} and \hbox{213} densities, and with fixed \hbox{123} and \hbox{321} densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.