Abstract
AbstractWe show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender 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.
Highlights
We will establish a connection between two classical objects: the Mallows measure on permutations and Gale–Shapley stable marriage. e Mallows measure Malnq on permutations of {, . . . , n} with parameter q ∈ [, ] is the probability measure that assigns to each permutation σ ∈ Sn a probability proportional to qinv(σ), where inv(σ) is the inversion number of σ, given by inv(σ) = (i, j) ∈ {, . . . , n} ∶ i < j but σ(i) > σ( j)
We define the Mallows measure MalIq on permutations of a general finite interval I ⊆ Z by shi ing the index. e Mallows measure was extended to permutations of infinite intervals by Gnedin and Olshanski [, ], who showed that for q ∈ [, ) and an infinite interval I ⊆ Z, the measures MalqI∩[−n,n] converge weakly to a probability measure MalIq on permutations of I
We call this limit the Mallows measure on permutations of I with parameter q. ey characterised the Mallows permutation of Z, together with its compositions with shi s, as the unique random permutations of Z with a property that they called q-exchangeability, which is equivalent to being a Gibbs
Summary
We show that, for both finite and infinite intervals, the Mallows permutation arises as a stable matching of the random bipartite graph on the interval. Every locally finite stable matching of KZ,Z(p) is perfect, and every stable matching of KZ,Z(p) is either tame or wild. We introduce a new algorithm for sequentially sampling the Mallows permutation that is well suited to studying cuts, and has a natural interpretation in terms of the matching. In this algorithm, an “alpha” male prevents less attractive males from finding partners until he himself finds one (at which point another male takes over as the alpha male). Denotes a discrete or continuous interval will always be clear from context
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
