First-order convergence for $321$-avoiding permutations

TL;DR

This paper proves that for large random 321-avoiding permutations, the probability of satisfying any first-order property converges, using advanced spectral methods, thus resolving an open problem in combinatorics.

Contribution

It establishes the first-order convergence law for 321-avoiding permutations, a significant open problem in the study of permutation classes.

Findings

Convergence law holds for random 321-avoiding permutations.

Proof uses an infinite-dimensional Perron-Frobenius theorem.

Resolves an open problem from recent combinatorics literature.

Abstract

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.