  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 fixed points of both 123-
and 231-avoiding permutations. We find an exact description for a scaling limit
of the empirical distribution of fixed points in term of Brownian excursion.
This builds on the connections between pattern-avoiding permutations and
Brownian excursion developed in Part I of this series and strengthens the
recent results of Elizalde (2012) and Miner and Pak (2014) on fixed points of
pattern-avoiding permutations.
