Mallows permutations as stable matchings

TL;DR

This paper reveals that Mallows permutations can be characterized as stable matchings in certain random bipartite graphs, extending the concept to infinite cases and classifying the types of stable matchings that occur.

Contribution

It establishes a novel connection between Mallows permutations and stable matchings in random bipartite graphs, including infinite cases, and classifies stable matchings into tame and wild types.

Findings

Mallows measure corresponds to the law of a unique stable matching in a random bipartite graph.

Almost surely, stable matchings in the infinite case fall into two classes: tame and wild.

Tame stable matchings are related to Mallows permutations of integers, with specific structural properties.

Abstract

We show that the Mallows measure on permutations of $1,\ldots,n$ arises as the law of the unique Gale-Shapley stable matching of the random bipartite graph conditioned to be perfect, where preferences arise from a total ordering of the vertices but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{\mathbb{Z},\mathbb{Z}}$ falls into one of two classes: a countable family $(\sigma_n)_{n\in\mathbb{Z}}$ of tame stable matchings, in which the length of the longest edge crossing $k$ is $O(\log |k|)$ as $k\to\pm \infty$, and an uncountable family of wild stable matchings, in which this length is $\exp \Omega(k)$ as $k\to +\infty$. The tame stable matching $\sigma_n$ has the law of the Mallows permutation of $\mathbb{Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$. The permutation $\sigma_{n+1}$ dominates $\sigma_{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.