  We initiate the study of limit shapes for random permutations avoiding a
given pattern. Specifically, for patterns of length 3, we obtain delicate
results on the asymptotics of distributions of positions of numbers in the
permutations. We view the permutations as 0-1 matrices to describe the
resulting asymptotics geometrically. We then apply our results to obtain a
number of results on distributions of permutation statistics.
