Scaling Limits for Width Two Partially Ordered Sets: The Incomparability   Window
  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. 
Nayantara Bhatnagar, Nick Crawford, Elchanan Mossel and Arnab Sen