The shape of a random pattern-avoiding permutation

A permutation that avoids the pattern 4321 has a longest decreasing sequence of length 3. We fix \$n\$, choose \$\sigma\$ a 4321-avoiding permutation uniformly at random and plot the points of the form \$(i/n,\sigma(i)/n)\$ for \$1 \leq i \leq n\$. Looking at this plot it is clear that the indices 1 through \$n\$ can be partitioned into three sets. By linear interpolation from these three sets we can generate three functions. We show that the scaling limit of this measure on triples of functions is given by the eigenvalues of a ensemble of random matrices. We also discuss the scaling limits of other patterns.