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 φ , the proportion of objects satisfying φ 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 [3] . Our proof relies on an infinite-dimensional version of the Perron-Frobenius theorem.