First-order convergence for $321$-avoiding permutations

We say that a convergence law holds for a sequence of random combinatorial objects if, for any first-order sentence $\varphi$, the proportion of objects satisfying $\varphi$ converges to a limiting value as the size of the objects tends to infinity. In this paper, we show that the convergence law holds for random $321$-avoiding permutations, settling an open problem posed in Albert, Bouvel, F\'eray, and Noy (2024). Our proof relies on an infinite-dimensional version of the Perron-Frobenius theorem.