The Shape of Random Pattern-Avoiding Permutations

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.