Mallows permutations as stable matchings

Abstract

We show that the Mallows measure on permutations of 1, . . . , 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 Z,Z falls into one of two classes: a countable family (σn) n∈Z of tame stable matchings, in which the length of the longest edge crossing k is O(log |k|) as k → ±∞, and an uncountable family of wild stable matchings, in which this length is exp Ω(k) as k → +∞. The tame stable matching σn has the law of the Mallows permutation of Z (as constructed by Gnedin and Olshanski) composed with the shift k → k + n. The permutation σn+1 dominates σn pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

Stationary distributions of the multi-type ASEP

“…Figure 1.2 shows two multi-line diagrams for the case N = 4, L = 10, and (k 1 , k 2 , k 3 , k 4 ) = (1,3,3,1). The sets of occupied sites are the same for the two diagrams.…”

“…The multi-type ASEP on Z also has stationary distributions which are not translationinvariant (whose projections onto one-type distributions are blocking measures in the sense of [9]). The connections between distributions of this type and the Mallows measure on permutations are studied by Gnedin and Olshanskii [22,23], and recent work by Angel, Holroyd, Hutchcroft and Levy [3] describes a link between such processes and a model of stable matchings.…”

“…Figure 1.2: Examples of multi-line diagrams, for N = 4, L = 10, and(k 1 , k 2 , k 3 , k 4 ) =(1,3,3,1). The configuration of occupied sites is the same in both cases.…”

Limit Distributions for Euclidean Random Permutations

“…Informally, one may think of a spatial random permutation as a random permutation which is biased towards the identity in some underlying geometry. This broad idea covers many specific cases including the well-studied interchange model [21,47,2,42,7,1,30,31,14,33,15] and the Mallows model (defined in dimension 1) [35,43,17,27,28,38,12,39,26,3]. The study of the cycle structure is of great interest in such models as well.…”

The height of Mallows trees

“…Mallows [22] in the context of ranking theory. The study of its probabilistic properties has taken off in the past decade; previously studied properties of Mallows permutations include the length of the longest increasing subsequence [4,6,23], the cycle structure [10,16,20,24,25], relations to exchangeability [17,18] and to random matchings [3], random dynamics with Mallows permutations as stationary distribution [5,12], and thermodynamic properties of Mallows measures [29,30].…”