Pattern-avoiding permutations and Brownian excursion Part I: Shapes and   fluctuations
  Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion. Exploring this connection to Brownian excursion allows us to strengthen the recent results of Madras and Pehlivan, and Miner and Pak as well as to understand many of the interesting phenomena that had previously gone unexplained. 
Christopher Hoffman, Douglas Rizzolo, and Erik Slivken
math.PR math.CO