Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

TL;DR

This paper investigates the asymptotic behavior of the number of incomparable elements in a random width-two partial order, revealing a connection to Brownian excursions and the Rayleigh distribution.

Contribution

It establishes a limit theorem linking the incomparability window in width-two partial orders to Brownian excursions, a novel probabilistic insight.

Findings

Number of incomparable elements converges to the height of a Brownian excursion.

Distribution of the incomparability window follows a Rayleigh distribution.

Provides a scaling limit for width-two partial orders.

Abstract

We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0,1], which follows the Rayleigh distribution.