  We say that a convergence law holds for a sequence of random combinatorial
objects if for any first-order sentence $\varphi,$ the density of the objects
that satisfy $\varphi$ converges to a limiting value. We show the convergence
law for random $321$-avoiding permutations, which was left as an open problem
in ABFN22. Our proof uses an infinite-dimensional version of the
Perron-Frobenius theorem.
