Mallows permutations as stable matchings
MMALLOWS PERMUTATIONS AS STABLE MATCHINGS
OMER ANGEL, ALEXANDER E. HOLROYD, TOM HUTCHCROFT, AND AVI LEVY
Abstract.
We show that the Mallows measure on permutations of 1 , . . . , n arises as the law ofthe 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 ismore intricate. We prove that almost surely every stable matching of the random bipartite graphobtained by performing Bernoulli percolation on the complete bipartite graph K Z , Z falls into one oftwo classes: a countable family ( σ n ) n ∈ Z of tame stable matchings, in which the length of the longestedge crossing k is O (log | k | ) as k → ±∞ , and an uncountable family of wild stable matchings, inwhich this length is exp Ω( k ) as k → + ∞ . The tame stable matching σ n has the law of the Mallowspermutation of Z (as constructed by Gnedin and Olshanski) composed with the shift k (cid:55)→ k + n .The permutation σ n +1 dominates σ n pointwise, and the two permutations are related by a shiftalong a random strictly increasing sequence. Introduction
In this paper we will establish a connection between two classical objects: the Mallows measureon permutations and Gale-Shapley stable marriage. The
Mallows measure
Mal nq on permutationsof { , . . . , n } with parameter q ∈ [0 ,
1] is the probability measure that assigns to each permutation σ ∈ S n a probability proportional to q inv( σ ) , where inv( σ ) is the inversion number of σ , given byinv( σ ) = (cid:8) ( i, j ) ∈ { , . . . , n } : i < j but σ ( i ) > σ ( j ) (cid:9) . More generally, we define the Mallows measure Mal Iq on permutations of a general finite interval I ⊆ Z by shifting the index. The Mallows measure was extended to permutations of infinite intervalsby Gnedin and Olshanski [10, 11], who showed that for q ∈ [0 ,
1) and an infinite interval I ⊆ Z ,the measures Mal I ∩ [ − n,n ] q converge weakly (with respect to the topology of pointwise convergence)to a probability measure Mal Iq on permutations of I . We call this limit the Mallows measure onpermutations of I with parameter q . They also characterised the Mallows permutation of Z , togetherwith its compositions with shifts, as the unique random permutations of Z with a property thatthey called q -exchangeability , which is equivalent to being a Gibbs measure on permutations of Z with respect to the Hamiltonian H ( σ ) = inv( σ ) and inverse temperature β = − log q .The Mallows measure was originally introduced in the context of statistical ranking theory [16].It has recently enjoyed substantial interest among both pure and applied mathematicians. In partic-ular, analysis has been carried out of the cycle structure [9] and the longest increasing subsequence[2, 4, 17] of a Mallows permutation, of the longest common subsequence of two independent Mallowspermutations [15], and of mixing times of related Markov chains [3, 6]. The Mallows permutationhas also been studied as a statistical physics model [18, 19], and has found applications in learning Date : 4 June 2018. a r X i v : . [ m a t h . P R ] J u l OMER ANGEL, ALEXANDER E. HOLROYD, TOM HUTCHCROFT, AND AVI LEVY theory [5] and in the theory of finitely dependent processes [13]. The Mallows measure also arisesas a stationary measure of the asymmetric exclusion process (ASEP) [1].In this paper, we show that, for both finite and infinite intervals, the Mallows permutationarises as a stable matching of the random bipartite graph on the interval. In particular, we obtaina new construction of the Mallows permutation of an infinite interval. The finite case followsin a straightforward way by consideration of known algorithms for sampling from the Mallowsdistribution, while the infinite case is more subtle and requires a more delicate treatment.The notion of stable matching was introduced in the hugely influential work of Gale and Shap-ley [8]. Since then, thousands of articles on the topic have been written, and Nobel Memorial Prizesin Economics have been awarded to Roth and Shapley for related work.Let us now describe informally the random stable matchings with which we shall be concerned.Suppose that we have a set of males and a set of females who seek to be matched into heterosexualpairs, in accordance with preferences defined as follows. Both the set of males and the set of femalesare ranked according to a universally agreed order of attractiveness. However, each male-femalepair has a probability q ∈ [0 ,
1) of being incompatible, independently of all other pairs, meaningthat neither will consider the other as a partner under any circumstances. Attractiveness andcompatibility are the only factors affecting preferences. In particular, if each set is finite, a female’sfirst choice for partner is the most attractive male she is compatible with, her second choice isthe second most attractive male she is compatible with, and so on. A matching is a collection ofcompatible pairs such that each individual is in at most one pair. A matching is stable if there doesnot exist a compatible male-female pair who would both prefer to be matched to each other overtheir current status, where any compatible partner is preferred to being unmatched.We now introduce notation in order to make these definitions more formal. A matching of agraph G is a set of edges no two of which share a vertex. A matching is perfect if every vertex has apartner. Given intervals I, J ⊆ Z , we write K I,J for the graph whose vertex set is ( I ×{ ♂ } ) ∪ ( J ×{ ♀ } )and whose edge set is (cid:8) { ( i, ♂ ) , ( j, ♀ ) } : i ∈ I, j ∈ J (cid:9) . If G is a subgraph of K I,J , we say that ( i, ♂ )and ( j, ♀ ) are compatible if there is an edge between them in G . Thus, a matching of K I,J is amatching of G if and only if every matched pair is compatible. We identify each matching of K I,J with a function σ : I → J ∪ {−∞} by setting σ ( i ) = j if ( i, ♂ ) is matched to ( j, ♀ ) and setting σ ( i ) = −∞ if ( i, ♂ ) is unmatched. The function σ − : J → I ∪ {−∞} is defined similarly byinterchanging the roles of ♂ and ♀ . In particular, if I = J and the matching is perfect then σ is apermutation of I and σ − is its inverse; this yields a bijection between perfect matchings of K I,I and permutations of I . A matching of a subgraph G of K I,J is stable if there does not exist a pair { ( i, ♂ ) , ( j, ♀ ) } such that ( i, ♂ ) is compatible with ( j, ♀ ), σ ( i ) < j , and σ − ( j ) < i .Given p ∈ [0 , K I,I ( p ) be the subgraph of K I,I with the same vertex set as K I,I andwhere each edge is included independently at random with probability p . Proposition 1.
Let p ∈ (0 , , let q = 1 − p , and let I ⊆ Z be an interval that is bounded above.Then the following hold. i. Every subgraph of K I,I has a unique stable matching, so that in particular K I,I ( p ) has aunique stable matching almost surely. ii. If I is finite then the unique stable matching of the random subgraph K I,I ( p ) is perfect withprobability (cid:81) | I | k =1 (1 − q k ) . iii. If I is infinite then the unique stable matching of K I,I ( p ) is perfect almost surely. ALLOWS PERMUTATIONS AS STABLE MATCHINGS 3
Figure 1.
Two realizations of the random bipartite graph K , (0 . Conditional on the event that the unique stable matching of K I,I ( p ) is perfect, it is distributedas a Mallows permutation of I with parameter q . We remark that the limit ( q ) ∞ := (cid:81) ∞ k =1 (1 − q k ) as | I | → ∞ of the probabilities appearing inProposition 1.ii is positive but strictly less than 1 for each q ∈ (0 , q ) ∞ := ∞ (cid:89) k =1 (1 − q k ) ∼ (cid:112) π (1 − q ) exp (cid:20) − π − q ) (cid:21) as q (cid:37) . The function ( q ) ∞ is the reciprocal of the generating function of integer partitions. It is alsoknown as both the q -Pochhammer symbol and the Euler function , owing to its role in Euler’spentagonal number theorem [7].The situation for intervals that are unbounded from above (so that there do not exist maximallyattractive individuals) in very different, and is the main topic of this paper. Indeed, for p ∈ (0 , K Z , Z ( p ) has uncountably many stable matchings, and even uncountably manystable matchings that are not perfect. We will prove, however, that the stable matchings of K Z , Z ( p )fall into two sharply distinguished classes: a countable family of tame matchings which correspondto compositions of the Mallows permutation of Z with shifts, and an uncountable family of wild stable matchings. Moreover, the tame and wild stable matchings have quantitatively very differentbehaviours. OMER ANGEL, ALEXANDER E. HOLROYD, TOM HUTCHCROFT, AND AVI LEVY
Figure 2.
A portion of the balanced tame stable matching of K Z , Z ( p ), for p = 0 . p = 0 . p = 0 . Z . Thin grey linesindicate edges whose endpoints must be incompatible for this matching to be stable.Other edges are omitted.To state these results we introduce some more definitions. For an interval I ⊆ Z , a matching σ of K I,I and i ∈ I , we define the quantities L + (cid:0) σ, i + (cid:1) = (cid:110) j ≤ i + such that σ ( j ) ≥ i + (cid:111) and L − (cid:0) σ, i + (cid:1) = (cid:110) j ≥ i + such that σ ( j ) ≤ i + (cid:111) . That is, L + ( σ, i + ) and L − ( σ, i + ) are the numbers of edges crossing over i + in each direction.We say that the matching σ is locally finite if L + ( σ, i + ) + L − ( σ, i + ) is finite for some (andhence every) i ∈ I . Note that matchings of intervals other than Z are always locally finite. If σ isperfect and locally finite, we define the flow of σ to beFl( σ ) = L + (cid:0) σ, i + (cid:1) − L − (cid:0) σ, i + (cid:1) , which is easily seen to be independent of i . We say that σ is balanced if it is perfect, locally finite,and has flow zero.For each matching σ of K Z , Z and each i ∈ Z , we also define M ( σ, i + ) = max (cid:110) | σ ( j ) − j | : j < i + < σ ( j ) or − ∞ < σ ( j ) < i + < j (cid:111) to be the length of the longest edge in the matching crossing i + , where the maximum of theempty set is taken to be zero. We say that a matching σ of K Z , Z is tame if it is locally finite andlim sup i →±∞ M (cid:0) σ, i + (cid:1) log | i | < ∞ , that is, if the maximum length of an edge of σ crossing i + is at most logarithmically large in i .On the other hand, we say that a matching σ is wild iflim inf i → + ∞ log M (cid:0) σ, i + (cid:1) i > ALLOWS PERMUTATIONS AS STABLE MATCHINGS 5
Figure 3.
Far left: the balanced tame matching σ of K Z , Z (0 . σ of flow 1 is obtained by composing σ with a shift by 1 (depicted as an upward shift of vertices on the right). Middle right:the matching σ for the same realization of the graph K Z , Z (0 . σ and σ superimposed. Edges unique to σ and σ are shown by thickblue and red lines respectively, while edges common to both matchings are shown bythin purple lines. The symmetric difference consists of a single bi-infinite path thatis increasing on both sides.that is, if the maximum length of an edge crossing i + grows at least exponentially as i → + ∞ . Inparticular, every matching that is not locally finite is wild. There is a wide gulf between tamenessand wildness – it is easy to construct matchings of K Z , Z that are neither tame nor wild. However,our stable matchings are either tame or wild. Theorem 2 (Tame/wild dichotomy) . Let p ∈ (0 , , let q = 1 − p , and consider the random bipartitegraph K Z , Z ( p ) . Almost surely, every locally finite stable matching of K Z , Z ( p ) is perfect, and everystable matching of K Z , Z ( p ) is either tame or wild. Simulated examples of tame and wild matchings of K Z , Z ( p ) are depicted in Figures 2 to 4. Notethat the definition of wildness is asymmetric, and does not say anything about the behaviouras i → −∞ . In fact, we will show that for every perfect, locally finite stable matching σ of K Z , Z ( p ), there is a tame stable matching that agrees with σ at all sufficiently large negative i –see Corollary 12. Next, we relate the tame matchings of K Z , Z ( p ) to the Mallows permutation, anddescribe some of their properties. Theorem 3 (Classification of tame matchings) . Let p ∈ (0 , , let q = 1 − p , and consider the randombipartite graph K Z , Z ( p ) . Almost surely, the tame stable matchings of K Z , Z ( p ) form a countable family ( σ n ) n ∈ Z with the following properties. i. For each n ∈ Z , the matching σ n is perfect and has flow n . ii. The stable matching σ n is the almost sure pointwise limit of the unique stable matching σ n,m of K ( −∞ ,m ] , ( −∞ ,m + n ] ( p ) as m → ∞ . iii. The matching σ n is distributed as the composition σ with the shift i (cid:55)→ i + n , where σ is aMallows-distributed permutation of Z with parameter q . OMER ANGEL, ALEXANDER E. HOLROYD, TOM HUTCHCROFT, AND AVI LEVY
Figure 4.
Portions of wild matchings of K Z , Z (0 . If n (cid:48) ≤ n , then the males do no better in σ n (cid:48) than in σ n , and the females do no worse.That is, σ n (cid:48) ( i ) ≤ σ n ( i ) and σ − n (cid:48) ≥ σ − n ( i ) for every i ∈ Z . Moreover, for every n ∈ Z , thematchings σ n and σ n +1 agree except on a strictly increasing sequence ( i n,k ) k ∈ Z , for which σ n ( i n,k ) = σ n +1 ( i n,k +1 ) for all k ∈ Z . In particular, note that while σ n +1 is distributed as the shift of σ n , it is not equal to the shift of σ n (see Figure 3). Theorem 3.iii is reminiscent of the situation for stable matchings with generalpreferences on finite sets, where there are two extremal matchings, one of which is female-optimaland male-pessimal while the other is male-optimal and female-pessimal [8].Finally, we prove that wild stable matchings do indeed exist. (The fact that tame matchings alsoexist is part of Theorem 3). Theorem 4 (Existence of wild matchings) . Let p ∈ (0 , , let q = 1 − p , and consider the ran-dom bipartite graph K Z , Z ( p ) . Almost surely, there exist stable matchings of K Z , Z ( p ) in each of thefollowing categories: • not perfect, • perfect but not locally finite, • perfect and locally finite, but wild.Indeed, there almost surely exist uncountably many stable matchings in each category. Also, if I is an infinite interval that is bounded from below, then K I,I ( p ) has uncountably many wild stablematchings almost surely. Matchings satisfying the conditions of Theorem 4 can be constructed via a simple and explicitalgorithm. (In particular, no appeal to the axiom of choice is required.)We remark that Theorem 2 and Theorem 4 are sharp in the sense that, disregarding constants, thedefinitions of tame and wild cannot be strengthened without the theorem becoming false. In otherwords, the tame stable matchings of K Z , Z ( p ) have logarithmically long edges, and there exist wild ALLOWS PERMUTATIONS AS STABLE MATCHINGS 7 stable perfect matchings of K Z , Z ( p ) in which the longest edge crossing i + is at most exponentiallylarge in i . See Propositions 13 and 16 respectively for the precise statements. About the proofs . Recall from Proposition 1 that there is a unique stable matching of everyinterval that is bounded from above. A central step in the proofs of Theorems 2 and 3 is to provethat the unique stable matchings of two different intervals, both bounded from above, coincide forall sufficiently large negative i , and moreover that the distance elapsed before they couple in thisway has an exponential tail. In particular, we will prove that if σ and σ (cid:48) are the unique stablematchings of K ( −∞ , , ( −∞ , ( p ) and K ( −∞ ,n ] , ( −∞ ,n ] ( p ) for some n ≥
0, then P (cid:16) σ ( − i ) = σ (cid:48) ( − i ) for all i ≥ k (cid:17) ≥ − (cid:2) − (1 − q )( q ) ∞ (cid:3) k . (1)Note that the bound on the right of (1) does not depend on n . The exact statement required forthe proofs of Theorems 2 and 3 is a little more general than this and is given in Proposition 7.To prove this proposition, we consider the mutual cuts of σ and σ (cid:48) . If σ is a permutation of Z ,we say that i + is a cut for σ if σ fixes the sets { j ∈ Z : j < i + } and { j ∈ Z : j > i + } . If, inthe setting above, i ≥ − i + is a cut for both σ and σ (cid:48) , then it follows from Proposition 1that σ ( − j ) = σ (cid:48) ( − j ) for all j ≥ i . Thus, to prove (1), it suffices to prove that (cid:16) (cid:2) − i + is a cut for both σ and σ (cid:48) (cid:3)(cid:17) i ≥ (2)stochastically dominates an i.i.d. Bernoulli process with parameter (1 − q )( q ) ∞ .The proof of Proposition 7 also yields the following variation of this result concerning a singlepermutation, which is of independent interest. Proposition 5.
Let q ∈ [0 , , let I be an infinite interval, and let σ be a random permutation of I drawn from the Mallows distribution with parameter q . Then the process (cid:16) (cid:2) i + is a cut for σ (cid:3)(cid:17) i ∈ Z stochastically dominates an i.i.d. Bernoulli process with parameter (1 − q )( q ) ∞ . We note that if σ is a Mallows permutation of Z then P ( i + is a cut for σ ) = ( q ) ∞ for every i ∈ Z ; this is an immediate consequence of Proposition 1 and Theorem 3, and is also aneasy consequence of the q -shuffling algorithm for sampling the Mallows permutation [11]. Thus, thedensity of the cuts of σ and of the Bernoulli process that Proposition 5 states that they dominatediffer by a factor of 1 − q .We now briefly discuss the proofs of Proposition 7 and Proposition 5. Proposition 1 naturallyleads to several algorithms for sequentially sampling the Mallows permutation, depending on theorder in which we choose to reveal the status of the edges in K I,I ( p ). Different algorithms lendthemselves to studying different aspects of the permutation. For example, the diagonal exposure algorithm of Gladkich and Peled [9], which is well suited to studying the cycle structure of theMallows permutation, is of this form. To prove Proposition 5, we introduce a new algorithm forsequentially sampling the Mallows permutation that is well suited to studying cuts, and has anatural interpretation in terms of the matching. In this algorithm, an “alpha” male prevents lessattractive males from finding partners until he himself finds one (at which point another male takes OMER ANGEL, ALEXANDER E. HOLROYD, TOM HUTCHCROFT, AND AVI LEVY
Figure 5.
Two matchings of equal flow agree at all sufficiently negative locations.Here p = 0 .
3, the two matchings have cuts at − and − respectively, and onlynegative integer locations are shown. Edges unique to one or other matching areshown by thick blue and red lines respectively, while edges common to both matchingsare shown by thin purple lines. The symmetric difference (restricted to negativelocations) consists of a single path with a unique locally minimal edge.over as the alpha male). A similar algorithm is implicit in the proof of [14, Proposition 8.1]. Toprove Proposition 7, we use a variation on this algorithm in which the two matchings σ and σ (cid:48) arecomputed simultaneously.2. Intervals with maximally attractive individuals
We now prove Proposition 1. As advertised in the introduction, we will also obtain a new proofthat the weak limit used to define the Mallows permutation of the infinite interval ( −∞ ,
0] exists.
Proof of Proposition 1.
Let I be an interval that is bounded from above. We may assume withoutloss of generality that max I = 0. Fix a subgraph G of K I,I . In any stable matching of G , themost attractive male must be matched to the most attractive female that is compatible with him.Inductively, the i th most attractive male must be matched to the most attractive female he iscompatible with among those who are not matched to a more attractive male. This shows that thestable matching is unique, and gives an algorithm to compute it. Formally, we set max ∅ = −∞ and define σ : I → I ∪ {−∞} recursively by setting σ (0) = max { k ∈ I : ( k, ♀ ) is compatible with (0 , ♂ ) } and, for all 1 ≤ i < | I | , σ ( − i ) = max (cid:16) { k ∈ I : ( k, ♀ ) is compatible with ( − i, ♂ ) } \ σ ([1 − i, (cid:17) , (3)where we use the notation σ ( A ) = { σ ( a ) : a ∈ A } . It follows by induction on i that σ is the uniquestable matching of G .Now suppose that G = K I,I ( p ). Then the probability that σ ( − i ) (cid:54) = −∞ given σ (0) , . . . , σ ( − i + 1)is equal to the probability that ( − i, ♂ ) is compatible with ( k, ♀ ) for some k in the set A i = I \ σ ([1 − i, . ALLOWS PERMUTATIONS AS STABLE MATCHINGS 9 If I is infinite, then A i is infinite for every i , and we deduce that σ ( − i ) (cid:54) = −∞ for every i ≥ | I | = n for some n ≥
0. In this case, on the event that σ ( − j ) (cid:54) = −∞ forall 0 ≤ j < i , the set A i has cardinality n − i . It follows that P (cid:0) σ ( − i ) (cid:54) = −∞ | σ (0) (cid:54) = −∞ , . . . , σ ( − i + 1) (cid:54) = −∞ (cid:1) = 1 − q n − i . Thus, the probability that the unique stable matching is perfect is given by P (cid:0) σ (0) (cid:54) = −∞ , . . . , σ ( − n + 1) (cid:54) = −∞ (cid:1) = n − (cid:89) i =0 (1 − q n − i ) = n (cid:89) i =1 (1 − q i ) . We next show that if I is finite, then the conditional distribution of the unique stable matching σ of K I,I ( p ) given that it is perfect is equal to Mal Iq . Fix a permutation τ of I . We wish to show thatthe probability that σ = τ is proportional to q inv( τ ) . By the recursive formula for σ given above, wehave that for all 1 ≤ i ≤ n , P (cid:0) σ ( − i ) = τ ( − i ) | σ (0) = τ (0) , . . . , σ ( − i + 1) = τ ( − i + 1) (cid:1) = (1 − q ) q { j< − τ ( − i ): − τ − ( − j ) >i } . Taking the product of these conditional probabilities and observing that n (cid:88) j =1 { j < − τ ( − i ) : − τ − ( − j ) > i } = inv( τ )yields that P ( σ = τ ) = q inv( τ ) (1 − q ) n as required. Note that this yields a proof of the well-knownformula (cid:88) τ ∈ S n q inv( τ ) = (cid:81) ni =1 (1 − q i )(1 − q ) n . Now suppose that I = ( −∞ , σ be the unique stable matching of K I,I ( p ), which isalmost surely perfect. It remains to prove that σ is the Mallows permutation of I as defined byGnedin and Olshanki [10, 11]. That is, we must prove that the law of σ is equal to the weak limitof the Mallows measures on permutations of I n = [ − n,
0] as n → ∞ , i.e., that P (cid:0) σ ( i ) = x i ∀ ≤ i ≤ k (cid:1) = lim n →∞ Mal I n q ( σ ( i ) = x i ∀ ≤ i ≤ k ) (4)for every k ≥ x , . . . , x k ∈ I . In fact, we will obtain as a corollary a new proof that this weaklimit exists, recovering the result of [10]. For each n ≥
0, let σ n be the unique stable matching of K I n ,I n ( p ) and observe that, by the above algorithm, σ n ( − i ) = (cid:40) σ ( − i ) if σ ( − i ) ≥ − n −∞ if σ ( − i ) < − n. Observe that for every n ≥
1, every k ≤ n and every x , . . . , x k ∈ I , we have that, by a similaranalysis to above, P (cid:0) σ n perfect | σ ( − i ) = x i ∀ ≤ i ≤ k (cid:1) = P (cid:0) σ n ( − j ) (cid:54) = −∞ ∀ ≤ j ≤ n | σ ( − i ) = x i ∀ ≤ i ≤ k (cid:1) = (cid:2) x i ∈ [ − n, ∀ ≤ i ≤ k (cid:3) k − n (cid:89) i =1 (1 − q i ) . Thus, we have thatMal I n q ( σ ( − i ) = x i ∀ ≤ i ≤ k ) = P ( σ ( − i ) = x i ∀ ≤ i ≤ k | σ n perfect)= (cid:2) x i ∈ [ − n, ∀ ≤ i ≤ k (cid:3) P ( σ ( i ) = x i ∀ ≤ i ≤ k ) (cid:81) n − ki =1 (1 − q i ) (cid:81) n +1 i =1 (1 − q i ) . The ratio of products at the end of the right-hand side tends to one as n → ∞ when k is fixed, andso we obtain that (4) holds as desired. (cid:3) Remark.
The proof of convergence shows that the restriction of σ n to [ − k,
0] is close to the infiniteMallows permutation not just for fixed k as needed, but even if k, n → ∞ jointly, as long as n − k → ∞ .3. Cuts, coupling, and the existence of the tame stable matchings
In this section we prove Proposition 7, below, which generalizes (1) from the introduction. Wethen use this proposition to prove the existential claims from Theorem 3. We begin by proving aspecial case of Proposition 5, applying to intervals that are bounded from above. Proposition 5will later follow by an easy limiting argument. Besides being of independent interest, the proof ofProposition 6 will serve as a warm-up to the proof of Proposition 7. We will then apply Proposi-tion 7 to prove Corollary 9 and Corollary 10, which establish the existential claims of Theorem 3.Proposition 7 will also be used in the following section to prove Theorem 2 and complete the proofof Theorem 3.
Proposition 6.
Let q ∈ [0 , , let I be an infinite interval that is bounded from above, and let σ bea random permutation of I drawn from the Mallows distribution with parameter q . Then the process (cid:16) (cid:2) i + is a cut for σ (cid:3)(cid:17) i ∈ I stochastically dominates an i.i.d. Bernoulli process with parameter (1 − q )( q ) ∞ . Before beginning the proof of Proposition 6, we note that, by re-indexing, Proposition 1 alsoimplies that for every p ∈ (0 ,
1] and every two non-empty sets
A, B ⊆ Z with | A | = | B | , both ofwhich are bounded above, there is a unique stable matching of K A,B ( p ), which we denote by σ A,B .Moreover, again by re-indexing and applying Proposition 1, the stable matching σ A,B is perfectalmost surely if A is infinite, and with probability (cid:81) | A | i =1 (1 − q i ) if A is finite. Proof.
We assume without loss of generality that I = ( −∞ , − σ n : ( −∞ , − → [ − n, − ∪ {−∞} , n ≥
1, by letting σ n ( − i ) = (cid:40) σ ( − i ) if i ≤ n and − σ ( − j ) ≤ n for all 1 ≤ j ≤ i −∞ otherwise . It suffices to prove that P (cid:0) − n − is a cut for σ | σ n (cid:1) ≥ (1 − q )( q ) ∞ . In fact we will prove the equality P (cid:0) − n − is a cut for σ | σ n (cid:1) = (1 − q ) U n (cid:89) k =1 (1 − q k ) , (5) ALLOWS PERMUTATIONS AS STABLE MATCHINGS 11 where U n is the number of males who are among the n most attractive (i.e., are in [ − n, − × { ♂ } )and are unmatched in the partial matching σ n . Note that − n − is a cut for σ if and only if U n +1 = 0.First, the most attractive male queries his compatibility with each of the n most attractivefemales, i.e., each female in [ − n, − × { ♀ } . If he finds he is not compatible with any of them, thenwe stop the procedure and do not match anyone. Otherwise, he is matched to the most attractive ofthese females with whom he is compatible. In this case, the second most attractive male queries hiscompatibility with each of the n most attractive females. If he is not compatible with any of thesefemales other than the one that is already matched, we stop the procedure and do not match anymales other than the most attractive one. Otherwise, we match the second most attractive male tothe most attractive of these females with whom he is compatible and who is not already matched.We continue this procedure recursively, finding matches for the males in order of attractivenessuntil we reach a male who cannot be matched, at which point we stop. Let F n be the σ -algebragenerated by all the information concerning compatibility that is revealed when computing σ n viathis procedure.Now suppose that we wish to compute σ n +1 , given σ n and the σ -algebra F n . We know thatthe most attractive male who is unmatched in σ n is not compatible with any of the females in[ − n, − × { ♀ } who are unmatched in σ n . Other than this, the only information we have aboutcompatibility concerns pairs of males and females at least one of whom is already matched in σ n ,and this information is no longer relevant for computing σ n +1 .Thus, to compute σ n +1 , we use the following procedure, illustrated in Figure 6. First, themost attractive male who is unmatched in σ n queries his compatibility with ( − n − , ♀ ), whois always unmatched in σ n . If he finds he is not compatible with her, we set σ n +1 = σ n andstop. Otherwise, he finds he is compatible with her. This occurs with probability p = 1 − q independently of everything that has happened previously. If this is the case, we then try to matchthe remaining U n unmatched males with the remaining U n unmatched females. Since no informationconcerning compatibility between any of these individuals has been revealed, we can re-index andapply Lemma 8 to deduce that the conditional probability that the stable matching between themis perfect is equal to (cid:81) | U n | k =1 (1 − q k ). It follows that P (cid:0) − n − is a cut for σ | σ n , F n (cid:1) = (1 − q ) U n (cid:89) k =1 (1 − q k ) , (6)and the equality (5) follows by taking the conditional expectation over F n given σ n . (cid:3) Remark.
In the sampling algorithm used above, the information concerning compatibilities that isrevealed when computing σ n +1 given σ n and F n is precisely F n +1 . In fact, ( U n ) n ≥ is a Markovchain with the filtration F n , and (5) gives the probability of jumping to 0. The associated Markovchain is positive recurrent by the above proposition, and a stationary Z -indexed Markov processwith the same transition rule can be used to sample the Mallows permutation of Z .We now come to the main technical result of this section. We say that a set A ⊆ Z is low if it is bounded from above and its complement Z \ A is bounded from below (equivalently, ifits symmetric difference with ( −∞ ,
0] is finite). For each low set A , we define r ( A ) to be thelargest integer such that ( −∞ , r ] ⊆ A . In particular, if σ is a locally finite matching then the set Figure 6.
Constructing the matching σ n +1 from σ n . (i) 4 males and 4 females areunmatched (orange dots). It is known that the most attractive unmatched “alpha”male is incompatible (shown as thin black lines) with all unmatched females. (ii) Anew male and female are revealed. (iii) The alpha male is found to be compatiblewith the new female, so they are matched and removed from consideration. (iv) Thenext most attractive unmatched male is found to be incompatible with the first andsecond most attractive females but compatible with the third, so they are matchedand removed from consideration. (v) The next most attractive unmatched male isfound to be incompatible with all unmatched females. He becomes the new alphamale and the step ends. Regardless of the initial state, the probability that the stepends with all the individuals matched is at least (1 − q )( q ) ∞ . σ (( −∞ , n ]) = { σ ( i ) : i ≤ n } is low for each n ∈ Z , with r (cid:0) { σ ( i ) : i ≤ n } (cid:1) = min i>n σ ( i ) − . By re-indexing, it follows from Proposition 1 that for any two low sets
A, B ⊆ Z , there is almostsurely a unique stable matching of K A,B ( p ), which we denote σ A,B , and which is perfect almostsurely. Since there are only countably many low sets, this holds for all low sets simultaneouslyalmost surely. We say that a pair of low sets A and B is balanced if | A \ B | = | B \ A | . Note thiscondition is equivalent to | A ∩ [ − n, ∞ ) | = | B ∩ [ − n, ∞ ) | for all sufficiently large n , or indeed for all n ≥ r ( A ) ∧ r ( B ). Proposition 7.
Let p = 1 − q ∈ (0 , and consider K Z , Z ( p ) . For (cid:96) ∈ { , } let ( A (cid:96) , B (cid:96) ) be a pair ofbalanced low subsets of Z , set r (cid:96) = r ( A (cid:96) ) ∧ r ( B (cid:96) ) , and set σ (cid:96) = σ A (cid:96) ,B (cid:96) , the unique stable matchingof K A (cid:96) ,B (cid:96) ( p ) . Then P (cid:0) σ ( − i ) = σ ( − i ) ∀ i ≥ max( n − r , n − r ) (cid:1) ≥ − (cid:104) − (1 − q )( q ) ∞ (cid:105) n for every n ≥ . In particular, σ ( − i ) = σ ( − i ) for all sufficiently large i almost surely. We remark that one may deduce a similar result (with a worse constant) from Proposition 5 viaa finite-energy argument.Our proof of Proposition 7 will use the following simple correlation inequality.
Lemma 8.
Let p = 1 − q ∈ (0 , . If A, B, A (cid:48) , B (cid:48) ⊆ Z are finite and non-empty with | A | = | B | and | A (cid:48) | = | B (cid:48) | , then P ( σ A,B and σ A (cid:48) ,B (cid:48) are both perfect ) ≥ | A | (cid:89) k =1 (1 − q k ) | A (cid:48) | (cid:89) k =1 (1 − q k ) . ALLOWS PERMUTATIONS AS STABLE MATCHINGS 13
In other words, the perfection of the stable matching of different pairs of sets are positivelycorrelated events. Note that these events are not increasing with respect to the compatibilitygraph, so that the claimed positive correlation does not follow from the FKG inequality.
Proof.
We prove the claim by induction on | A | + | A (cid:48) | . The cases in which either | A | = 1 or | A (cid:48) | = 1,and in particular the case | A | + | A (cid:48) | = 2, are trivial. Thus, suppose that | A | , | A (cid:48) | ≥
2, and that theclaim has been proven for all pairs of pairs of finite non-empty sets ˜ A, ˜ B, ˜ A (cid:48) , ˜ B (cid:48) ⊆ Z with | ˜ A | = | ˜ B | ,and | ˜ A (cid:48) | = | ˜ B (cid:48) | , and | ˜ A | + | ˜ A (cid:48) | < | A | + | A (cid:48) | . Let i = max A ∪ A (cid:48) , so that ( i, ♂ ) is the most attractivemale in ( A ∪ A (cid:48) ) × { ♂ } .First suppose that i is in exactly one of A or A (cid:48) ; without loss of generality we may assume that i ∈ A \ A (cid:48) . As in the proof of Proposition 1, the male ( i, ♂ ) must be matched in σ A,B to the mostattractive female he is compatible with in B × { ♀ } , and the probability that there is at least onesuch compatible female is 1 − q | A | . Note that on the event that ( i, ♂ ) is matched to ( j, ♀ ) in σ A,B we have that σ A,B ( k ) = σ A \{ i } ,B \{ j } ( k ) for every k ∈ A \ { i } . Note also that the only informationrequired to compute σ A,B ( i ) concerns compatibility information between ( i, ♂ ) and Z × { ♀ } , andthat, given σ A,B ( i ) = j , this information is no longer relevant for computing either σ A \{ i } ,B \{ j } or σ A (cid:48) ,B (cid:48) . Thus, it follows by the induction hypothesis that for every j ∈ B , P (cid:0) σ A,B and σ A (cid:48) ,B (cid:48) are both perfect | σ A,B ( i ) = j (cid:1) = P (cid:0) σ A \{ j } ,B \{ i } and σ A (cid:48) ,B (cid:48) are both perfect (cid:1) ≥ | A |− (cid:89) k =1 (1 − q k ) | A (cid:48) | (cid:89) k =1 (1 − q k ) . The result now follows since P ( σ A,B ( i ) = j for some j ∈ B ) = 1 − q | A | .Now suppose that i ∈ A ∩ A (cid:48) . In order for σ A,B and σ A (cid:48) ,B (cid:48) both to be perfect, ( i, ♂ ) must becompatible with both a female from B × { ♀ } and a female from B (cid:48) × { ♀ } , with these two femalespossibly being the same. The probability of the required females existing is1 − q | B | − q | B (cid:48) | + q | B ∪ B (cid:48) | ≥ (1 − q | B | )(1 − q | B (cid:48) | ) . (7)Arguing similarly to the previous case, we have that P (cid:0) σ A,B and σ A (cid:48) ,B (cid:48) are both perfect | σ A,B ( i ) = j, σ A (cid:48) ,B (cid:48) ( i ) = j (cid:48) (cid:1) = P (cid:0) σ A \{ j } ,B \{ i } and σ A (cid:48) \{ i } ,B (cid:48) \{ j (cid:48) } are both perfect (cid:1) for every j ∈ B and j (cid:48) ∈ B (cid:48) such that the event being conditioned on has positive probability, andthe claim follows from the induction hypothesis together with (7). (cid:3) We remark that the same argument also yields analogous inequalities for more than two pairs.
Proof of Proposition 7.
We may assume without loss of generality that r ∧ r = 0. The proof isan elaboration of the proof of Proposition 6, and we omit some minor details. First, observe thatif i ≥ − i + is a cut for both σ and σ , then the restrictions of σ and σ to ( −∞ , − i ]both define stable matchings of K ( −∞ , − i ] , ( −∞ , − i ] ( p ), and hence are equal by Proposition 1. Thus, itsuffices to prove that P (cid:0) ∃ i ∈ [1 , n ] such that − i + is a cut for both σ and σ (cid:1) ≥ − (cid:104) − (1 − q )( q ) ∞ (cid:105) n for all n ≥ We perform a similar sampling procedure to that used in the proof of Proposition 5. For both (cid:96) ∈ { , } , we define a sequence of partial matchings σ (cid:96),n : A (cid:96) → B (cid:96) ∩ [ − n, ∞ ) ∪ {−∞} for n ≥ − σ (cid:96),n ( i ) = σ (cid:96) ( i ) if the following three conditions hold for i ∈ A (cid:96) , and otherwise σ (cid:96),n ( i ) = −∞ . • i ≥ − n • σ (cid:96) ( j ) ≥ − n for all j ∈ A (cid:96) ∩ [ i, ∞ ) • σ − (cid:96) ( j ) ≥ − n for all j ∈ A − (cid:96) ∩ ( i, ∞ )It suffices to prove that P (cid:0) − n − is a cut for both σ and σ | σ ,n , σ ,n (cid:1) ≥ (1 − q )( q ) ∞ for every n ≥ −
1. We will prove the stronger bound P (cid:0) − n − is a cut for both σ and σ | σ ,n , σ ,n (cid:1) ≥ (1 − q ) U ,n +1 (cid:89) k =1 (1 − q k ) U ,n +1 (cid:89) k =1 (1 − q k ) (8)where U (cid:96),n is the number of males in ([ − n, ∞ ) ∩ A ) × { ♂ } that are unmatched in σ (cid:96),n . (Unlike inthe proof of Proposition 5, this is not an equality in general.)Similarly to the proof of Proposition 5, we try to match the males in ( A ∪ A ) × { ♂ } in orderof attractiveness. At each step, we may need to find them a match in either σ ,n , σ ,n , or both,according to whether they are in A × { ♂ } , A × { ♂ } , or both. If we reach a male for whom wecannot find both the required matches, we stop. If that male can be matched in one of σ ,n , σ ,n butnot the other, we make the single match, and stop. During this procedure, the information revealedcan be described as follows: • Consider the most attractive male in ( A ∪ A ) × { ♂ } that is unmatched either in σ ,n , in σ ,n , or in both. For (cid:96) = 1 or 2, if this male is unmatched in σ (cid:96),n , then we know that he isnot compatible with any female in ([ − n, ∞ ) ∩ B (cid:96) ) × { ♀ } who is unmatched in σ (cid:96),n . • Other than this, the only information we have about compatibility concerns pairs of malesand females such that for each (cid:96) = 1 ,
2, at least one of the pair is either already matchedin σ (cid:96),n or is not in the set ( A (cid:96) × { ♂ } ) ∪ ( B (cid:96) × { ♀ } ). The status of these edges is no longerrelevant for computing σ ,n +1 and σ ,n +1 .Thus, to compute σ ,n +1 and σ ,n +1 given σ ,n and σ ,n , we may do the following. First, the mostattractive male who is unmatched in either σ ,n or σ ,n queries his compatibility with ( − n − , ♀ ). Ifhe finds he is not compatible with her, we set σ ,n +1 = σ ,n and σ ,n +1 = σ ,n and stop. Otherwise,he finds he is compatible with her. This occurs with probability p = 1 − q independently of everythingthat has happened previously. If this is the case, we match him to her in whichever of the matchings σ ,n and/or σ ,n that he was previously unmatched in. Call these updated matchings σ (cid:48) ,n and σ (cid:48) ,n .At this point, the number of remaining unmatched males (and females) in σ (cid:48) (cid:96),n is either U (cid:96),n or U (cid:96),n + 1, and no information about their compatibility has been revealed so far. Thus, by Lemma 8,the conditional probability that all the unmatched individuals in σ (cid:48) ,n and in σ (cid:48) ,n both support aperfect stable matching is at least U ,n +1 (cid:89) k =1 (1 − q k ) U ,n +1 (cid:89) k =1 (1 − q k ) ≥ ( q ) ∞ . Thus the conditional probability that − n − is a cut for both σ , σ is at least (1 − q )( q ) ∞ , andthe claim follows. ALLOWS PERMUTATIONS AS STABLE MATCHINGS 15
The final part of the proposition follows by continuity of measure. (cid:3)
Corollary 9 (Existence of limit matchings) . Let p ∈ (0 , . Let n ∈ Z , and for each m ≥ let σ n,m be the unique stable matching of K ( −∞ ,m + n ] , ( −∞ ,m ] ( p ) . Then σ m,n converges almost surely as m → ∞ to a permutation σ n of Z , and the limit is distributed as the composition of the Mallowspermutation of Z with the shift θ n : i (cid:55)→ i + n .Proof. It suffices to consider the case n = 0, as the others then follow by re-indexing. We write σ m = σ ,m . For each k ∈ Z , it follows from Proposition 7 that (cid:88) m>k P (cid:16) σ m ( k ) (cid:54) = σ m +1 ( k ) (cid:17) < ∞ . It follows by Borel-Cantelli that σ m almost surely converges pointwise to a function σ : Z → Z . Bysymmetry, the inverse functions σ − m also almost surely converge pointwise to a function σ − : Z → Z .Since σ − m ◦ σ m ( i ) = σ m ◦ σ − m ( i ) = i for every i ∈ ( −∞ , m ], it follows that σ − ( σ ( i )) = σ ( σ − ( i )) = i for every i ∈ Z , so that σ is a permutation almost surely. The fact that σ is Mallows-distributedwith parameter q , and in particular that the limit defining this permutation exists, follows from thecorresponding statement for σ m , proven in Proposition 1. (cid:3) Proof of Proposition 5.
This follows from Proposition 6 and Corollary 9, by taking the result throughto the limit. (cid:3)
We call the matching σ n from Corollary 9 the Mallows matching of K Z , Z ( p ) with flow n . Corollary 10 (Mallows matchings are tame) . Let q ∈ [0 , , and let σ be a random permutation of Z drawn from the Mallows distribution with parameter q . Then almost surely the composition of σ with the shift θ n : i (cid:55)→ i + n is tame for all n .Proof. Tameness is clearly invariant to composition with a shift, so it suffices to consider the case n = 0. Observe that if ( T i + ) i ∈ Z are the sequence of cuts of σ in order, then for every i ∈ Z max (cid:8) M ( σ, j + ) : T i ≤ j ≤ T i +1 (cid:9) ≤ | T i +1 − T i | . Thus, if σ is the Mallows permutation of Z , then it follows from Proposition 5 that P (cid:0) M ( σ, i + ) ≥ m (cid:1) ≤ c ( q ) m for some c ( q ) <
1. It follows by Borel-Cantelli that M ( σ, i + ) = O (log | i | ) almost surely, so that σ is tame as claimed. (cid:3) We also have the following immediate corollaries of Proposition 7, showing that every perfectlocally finite matching is “tame towards −∞ ”. Corollary 11.
Let p ∈ (0 , and consider K Z , Z ( p ) . Almost surely, every perfect, locally finite,balanced stable matching σ of K Z , Z ( p ) has a cut. Corollary 12.
Let p ∈ (0 , and consider K Z , Z ( p ) . Then the following holds almost surely. Forevery perfect, locally finite stable matching σ of K Z , Z ( p ) with flow n , there exists N σ such that σ ( − i ) = σ n ( − i ) for every i ≥ N σ , where σ n is the Mallows matching of K Z , Z ( p ) with flow n . Finally, let us note that Theorem 2 is sharp in the sense that, disregarding constants, the definitionof tameness cannot be strengthened.
Proposition 13.
Let p = 1 − q ∈ (0 , and let σ be the balanced Mallows matching of K Z , Z ( p ) .Then lim inf i →∞ M (cid:0) σ, i + (cid:1) log | i | ≥
12 log q − almost surely.Proof. For each i ∈ Z , let X i = min {| i − j | : ( j, ♀ ) is compatible with ( i, ♂ ) } . Then the randomvariables X i are i.i.d. with P ( X i ≥ n ) = q n − for all n ≥
1. Thus, for all a > i ≥ e /a we have that P ( X i ≥ a log i ) = q (cid:98) a log i (cid:99)− . When a − = 2 log q − , the latter expression has infinite sum (over i ≥ e /a ) and thus by the Borel-Cantelli Lemma X i ≥ a log | i | infinitely often, almost surely. The claim follows since M ( σ, i + ) ∧ M ( σ, i − ) ≥ X i . (cid:3) The dichotomy between tame and wild
In this section we complete the proof of Theorems 2 and 3. The central additional ingredientsrequired are the following two lemmas concerned with perfect, locally finite stable matchings.
Lemma 14.
Let p ∈ (0 , , let q = 1 − p , and consider the random bipartite graph K Z , Z ( p ) . Thereexists a positive constant c ( q ) such that the following holds almost surely: Every perfect, locallyfinite stable matching σ of K Z , Z ( p ) satisfying lim inf i → + ∞ i M (cid:0) σ, i + (cid:1) ≤ c ( q ) (9) is equal to the Mallows stable matching of K Z , Z ( p ) with the same flow as σ . In particular, σ is tameif and only if (9) holds. Lemma 15.
Let p ∈ (0 , , let q = 1 − p , and consider the random bipartite graph K Z , Z ( p ) . Almostsurely, every perfect, locally finite stable matching σ of K Z , Z ( p ) satisfying lim inf i → + ∞ i M ( σ, i + ) > is wild. In particular, σ is wild if and only if (10) holds. Before proving these lemmas, let us use them to prove Theorem 2 and Theorem 3.
Proof of Theorem 2.
Let Ω be the almost sure event that for every i ∈ Z , the sets { j : ( j, ♀ )compatible with ( i, ♂ ) } and { j : ( j, ♂ ) compatible with ( i, ♀ ) } are both unbounded below. Weclaim that every locally finite stable matching of K Z , Z ( p ) is perfect on the event Ω. Indeed, supposewithout loss of generality that ( i, ♂ ) is unmatched in some stable matching σ of K Z , Z ( p ). Then eachelement of the set { j : ( j, ♀ ) compatible with ( i, ♂ ) } must be matched to a male more attractive than( i, ♂ ), and thus σ is not locally finite. The remaining claims of the theorem follow from Lemmas 14and 15, since at least one of (9) or (10) must hold for every σ . (cid:3) Proof of Theorem 3.
The fact that a tame stable matching of flow n exists almost surely for each n follows from follows from Corollaries 9 and 10, while the fact that the matching is unique and canbe described as the limit of the unique stable matchings σ n,m of K ( −∞ ,m ] , ( −∞ ,m + n ] ( p ) as m → ∞ follows from Lemma 14. Given this limiting construction, Theorem 3.iii follows from Proposition 1. ALLOWS PERMUTATIONS AS STABLE MATCHINGS 17
For Theorem 3.iv, fix m ≥ n ∈ Z . Let ( i k ) k ≥ and ( j k ) k ≥ be decreasing sequences definedrecursively by i = m + n + 1, j k = σ − m,n +1 ( i k − ) for every k ≥ i k = σ m,n ( j k ) for every k ≥
1. Considering the gender-reversal of the recursive procedure for computing σ m,n and σ m,n +1 as in the proof of Proposition 1, beginning by assigning matches to the most attractive females in( −∞ , m + n ] and ( −∞ , m + n + 1] respectively, it follows by a straightforward induction argumentthat σ − m,n +1 ( i ) = (cid:40) j k +1 if i = i k for some k ≥ ,σ − m,n ( i ) otherwiseand similarly that σ m,n ( j ) = (cid:40) i k if j = j k for some k ≥ σ m,n +1 ( i ) otherwise.The second part of Theorem 3.iv follows by taking the limit as m → ∞ . The first part follows fromthe second, since if for each n males do no worse in σ n +1 than in σ n , then the same comparisonholds for any σ n , σ n (cid:48) . (cid:3) Proof of Lemma 14.
By re-indexing, it suffices to consider the balanced case of flow 0. Let 0 < a < A i,a be the set of low subsets A of Z such that the pair of low sets (( −∞ , i ] , A ) is balanced, andthe symmetric difference of A with ( −∞ , i ] is contained in [(1 − a ) i, (1 + a ) i ]. Clearly | A i,a | ≤ ai +1 .Let k ≥
0. By Proposition 7, for every A ∈ A i,a we have that P (cid:16) σ ( −∞ ,i ] ,A ( j ) = σ ( −∞ ,i ] , ( −∞ ,i ] ( j ) for every j ≤ k (cid:17) ≥ − c ( q ) (1 − a ) i − k , where c ( q ) = 1 − (1 − q )( q ) ∞ . Thus, it follows by the union bound that P (cid:0) σ ( −∞ ,i ] ,A ( j ) = σ ( −∞ ,i ] , ( −∞ ,i ] ( j ) for every j ≤ k and A ∈ A i,a (cid:1) ≥ − ai +1 c ( q ) (1 − a ) i − k . Fix a > a c ( q ) (1 − a ) < a exists since c ( q ) < a := ∀ k ∃ (cid:96) ∀ i ≥ (cid:96) (cid:8) σ ( −∞ ,i ] ,A ( j ) = σ ( −∞ ,i ] , ( −∞ ,i ] ( j ) for every j ≤ k and A ∈ A i,a (cid:9) occurs almost surely. Consider the set of balanced, perfect, locally finite stable matchings σ of K Z , Z ( p ) that satisfy lim inf i → + ∞ M (cid:0) σ, i + (cid:1) i < a. For any such matching, σ (( −∞ , i ]) belongs to A i,a for infinitely many i . However, on the event Ω a ,the balanced Mallows stable matching of K Z , Z ( p ) is the only stable matching of K Z , Z ( p ) with thisproperty, and so σ must be equal to this matching. (cid:3) Proof of Lemma 15.
By re-indexing it suffices to consider balanced stable matchings. Moreover,since by Corollary 11 there is a probability one event on which every perfect, locally finite, balancedstable matching of K Z , Z ( p ) has a cut, it suffices by re-indexing to consider matchings that have acut at . That is, it suffices to prove that there is a probability one event on which every perfect,locally finite, balanced stable matching σ of K Z , Z ( p ) that has a cut at 1 / i → + ∞ i M ( σ, i + ) > i (1 + (cid:15) ) ie δi Figure 7.
Exponential blow-up in wild matchings. Edges whose length is linear intheir height necessitate edges whose length is exponential in their height (with highprobability as the height goes to infinity).is wild.For ε, δ >
0, define Ω ε,δ to be the event on which there exists
N < ∞ such that the followingconditions hold for all i ≥ N .(i) The inequalities (cid:8) i ≤ j ≤ i + εi : ( j, ♂ ) is compatible with ( i + k, ♀ ) (cid:9) ≥ − q εi and (cid:8) i ≤ j ≤ i + εi : ( j, ♀ ) is compatible with ( i + k, ♂ ) (cid:9) ≥ − q εi both hold for every 1 ≤ k ≤ e δi .(ii) There do not exist sets A, B ⊆ [0 , e δi ] with | A | , | B | ≥ − q εi such that no male in A × { ♂ } is compatible with any female in B × { ♀ } .We claim that for each ε >
0, there exists δ ( ε, q ) > ε,δ occurs almost surely. Indeed,for condition (i), the Chernoff bound implies that there exists a constant c ( ε, q ) < i, k ≥
0, the probability that either of the sets in question is smaller than (1 − q ) εi/ c ( ε, q ) i . Thus, summing over the possible choices of k and i and applying Borel-Cantelli shows thatif δ is sufficiently small then the required inequalities hold for all i sufficiently large almost surely.For condition (ii), it suffices to consider A, B of the minimal possible size s = − q εi . Countingthe choices of A and B and using the the union bound gives the following upper bound on theprobability that there exist sets A, B violating (ii): P (cid:0) there exist such A, B ⊆ [0 , e δi ] (cid:1) ≤ (cid:18) e δi s (cid:19) q s ≤ e δis − s | log q | = e C ( ε,δ ) i , ALLOWS PERMUTATIONS AS STABLE MATCHINGS 19 where we have used the elementary inequality (cid:0) mn (cid:1) ≤ m n . If δ is sufficiently small then C ( ε, δ ) < (cid:84) n ≥ Ω /n,δ (1 /n ) occurs. Let σ be a stable perfectmatching of K Z , Z ( p ) that is balanced and has a cut at , let n ≥ ε = 1 /n < lim inf M ( σ, i + 1 / /i , and let δ = δ ( ε, q ). For each a, b ≥
0, we define the indicator functions F ( a, b ) = (cid:0) ∃ i ≤ a such that σ ( i ) > b or σ − ( i ) > b (cid:1) . We claim that if i ≥ N and F ( i, i + εi ) = 1, then we must have that F ( i + εi, e δi ) = 1 also. Giventhis, it is easily seen that σ is wild: Indeed, our choice of ε guarantees that for every sufficiently large i there is an edge of length at least 2 εi spanning i , so that at least one of F ( i, i + εi ) or F ( i − εi, i )is equal to 1 for every sufficiently large i . In the first case we have that F ( i + εi, e δi ) = 1 and inthe second we have that F ( i, e δ ( i − εi ) ) = 1. In either case we deduce that M ( σ, i + 1 / ≥ e δi/ ,concluding the proof.To prove the claim above, let k = (cid:100) εi (cid:101) , and suppose that F ( i, i + k ) = 1. By symmetry betweenmales and females, let us suppose without loss of generality that some j ≤ i has σ ( j ) > i + k . If σ ( j ) > e δi , we are done. Otherwise, let V be the set of males in ( i, i + k ] × { ♂ } that are compatiblewith σ ( j ). By (i) we have that | V | ≥ − q k . Note that each male ( a, ♂ ) ∈ V must have σ ( a ) > σ ( j ),since otherwise σ would be unstable. Moreover, if any a ∈ V has σ ( a ) > e δi we are done, so we maysuppose not.Since σ is balanced and has a cut at 1 /
2, there is a set W ⊆ [1 , i + k ] of size at least k such that σ − ( w ) > i for every w ∈ W . If any w ∈ W has σ − ( w ) > e δi , we are done, so suppose not. Sinceeach female in W × { ♀ } is less attractive than σ ( j ) and is matched to a male more attractive than( j, ♂ ), stability implies that they cannot be matched to any male in V × { ♂ } . It follows that thereis a set W (cid:48) ⊂ W of females of size at least | V | that are matched to males in ( i + k, e δi ] × { ♂ } .Let B = σ ( V ) and A = σ − ( W (cid:48) ). Summarizing our conclusions, V, W (cid:48) are sets of size at least − q k in [0 , i + k ], and A, B are sets of size at least − q k in ( i + k, e δi ]. Stability implies that no malein A × { ♂ } is compatible with any female in B × { ♀ } , which contradicts (ii). (cid:3) Remark.
It is possible to show using the above proof that for every p = 1 − q ∈ (0 , c ( q ) such thatlim inf i → + ∞ log M (cid:0) σ, i + (cid:1) i ≥ c ( q )for every wild stable matching of K Z , Z ( p ) almost surely.5. Existence of wild stable matchings
Proof of Theorem 4.
For a finite set A ⊂ Z and for an integer i ∈ Z \ A , let Ω A,i be the event thatboth of the sets (cid:8) j ∈ Z : ( j, ♀ ) is compatible with ( i, ♂ ) but not ( k, ♂ ) for any k ∈ A (cid:9) and (cid:8) j ∈ Z : ( j, ♂ ) is compatible with ( i, ♀ ) but not ( k, ♀ ) for any k ∈ A (cid:9) are unbounded from above. Clearly P (Ω A,i ) = 1 for all A and i . Since there are only a countablenumber of finite subsets of Z , the eventΩ = (cid:92) { Ω A,i : A finite subset of Z , i ∈ Z \ A } also has probability 1.We will show that the conclusions of the theorem hold on Ω. All three claims will follow fromvariants of the same construction. Let ( x n ) n ≥ be an enumeration of a subset of Z × { ♂ , ♀ } (rep-resenting individuals to be matched in a given order), and let ( a j ) j ≥ be a sequence of positiveintegers (which will encode uncountably many options). We take the enumerated subset to be oneof the following:(1) (cid:0) Z × { ♂ } (cid:1) ∪ (cid:0) [0 , ∞ ) × { ♀ } (cid:1) (2) Z × { ♂ , ♀ } .(3) [0 , ∞ ) × { ♂ , ♀ } .These three choices are used to construct matchings that are respectively: not perfect; perfect butnot locally finite; and perfect, locally finite, and wild. For each fixed sequence ( x n ) n ≥ , every choiceof the sequence ( a j ) j ≥ will yield a different stable matching with this desired properties.At each step of the construction, we choose the next individual in the sequence, and if they arenot already matched, we find a compatible partner for them who is incompatible with everyonepreviously matched. At the end of step n ≥ σ n with the property that the vertices x , . . . , x n are all matched, and that σ n is a stable perfectmatching of the subgraph induced by the set of all matched vertices.Initially we take σ to be the empty matching. Given σ n for some n ≥
0, we define σ n +1 asfollows. • If x n +1 is already matched in σ n then we set σ n +1 = σ n . (That is, we do nothing.) • If x n +1 is not matched in σ n , then we choose a partner for it as follows. Suppose x n +1 isa female (the case of a male x n +1 is similar). We will match ( x n +1 , ♀ ) to a male in the set S × { ♂ } , where S = k ≥ k, ♂ ) is more attractive than any male that has already beenmatched, is compatible with x n +1 , and is not compatible with anyfemale that is matched in σ n . (We take k ≥ S is bounded from below.) On Ω, the set S is unbounded fromabove. If σ n has j edges, we match ( x n +1 , ♀ ) to the a j +1 -th least attractive male in S × { ♂ } .Let σ = lim σ n . As a result, the following hold. • If ( x i ) i ≥ is an enumeration of A = ( Z ×{ ♂ } ) ∪ ([0 , ∞ ) ×{ ♀ } ), then σ is a perfect matching of A , with ( −∞ , − × { ♀ } left unmatched. Such a matching must also be a stable matching of K Z , Z ( p ), since every male in the matching prefers their partner to every female in ( −∞ , − • If ( x i ) i ≥ is an enumeration of Z × { ♂ , ♀ } , taking the limit as n → ∞ we obtain a perfectstable matching of K Z , Z ( p ). The construction ensures that for every k >
0, the partner of( − k, ♀ ) belongs to [0 , ∞ ) × { ♂ } , so that σ is not locally finite. • If ( x i ) i ≥ is an enumeration of [0 , ∞ ) × { ♂ , ♀ } , then σ is a stable perfect matching of K [0 , ∞ ) , [0 , ∞ ) ( p ). Combining this with the unique stable matching of K ( −∞ , − , ( −∞ , − ( p ), weobtain a stable perfect matching of K Z , Z ( p ) with a cut at − , which must be balanced. ALLOWS PERMUTATIONS AS STABLE MATCHINGS 21
It is clear that, in each case, different sequences ( a n ) yield different matchings σ . Indeed, atthe first place two sequences differ, the edge added to the matchings will also differ. In the lastcase (locally finite matchings) we have not ruled out that σ is the unique balanced, tame stablematching. However, excluding this matching still leaves uncountably many locally finite, balanced,wild stable matchings. Similar statements for other values of the flow follow by re-indexing.Finally, note that in the third case, the restrictions to K [0 , ∞ ) , [0 , ∞ ) ( p ) of the matchings that weobtain are stable and perfect, and by re-indexing we obtain that K I,I ( p ) has uncountably manywild stable matchings almost surely for every infinite interval I that is bounded from below. (cid:3) Finally, we show that Theorem 2 is sharp in the sense that, disregarding constants, the definitionof wildness cannot be strengthened.
Proposition 16.
Let p = 1 − q ∈ (0 , and consider the random bipartite graph K Z , Z ( p ) . Thenthere almost surely exists a perfect, locally finite, wild stable matching of K Z , Z ( p ) such that lim sup i → + ∞ log M (cid:0) σ, i + (cid:1) i ≤ q − . (12) Proof.
Let ( x i ) i ≥ be the enumeration of [1 , ∞ ) × { ♂ , ♀ } given by x i − = ( i, ♂ ) and x i = ( i, ♀ ) forevery i ≥
1. As in the proof of Theorem 4, we define a sequence of partial matchings ( σ n ) n ≥ asfollows. Let σ be the unique stable perfect matching of K ( −∞ , , ( −∞ , . Having defined σ n , if x n +1 is matched in σ n then σ n +1 = σ n . If x n +1 is not matched in σ n , let σ n +1 be obtained from σ n bymatching x n to the least attractive member of the opposite gender that is compatible with x n +1 ,is strictly more attractive than every individual (of either gender) that is already matched in σ n ,and is incompatible with every individual of the same gender as x n +1 that has positive index andis already matched in σ n . As before, we obtain a locally finite stable perfect matching σ of K Z , Z ( p )by taking the limit as n → ∞ .We wish to verify that this matching σ satisfies (12). For each n ≥
0, let H n be the index of themost attractive individual (of either gender) that is already matched in σ n . For each n ≥
1, let N n be the number of individuals of positive index that are of the same gender as x n and already matchedin σ n − , and note that (cid:98) ( n − / (cid:99) ≤ N n ≤ n −
1. Then, conditional on σ n , we either have that x n +1 is already matched in σ n in which case H n +1 − H n = 0, or else H n +1 − H n − pq n ≤ pq N n +1 ≤ pq (cid:98) n/ (cid:99) . Inparticular, it follows that H n +1 − H n ≥ p − q −(cid:98) n/ (cid:99) for infinitely many n almost surely, which clearlyimplies that σ is not tame and is therefore wild almost surely by Theorem 2. On the other hand, italso follows that E [ H n ] ≤ n − (cid:88) i =0 (cid:16) pq i (cid:17) = q − n + o ( n ) , and it follows by an easy application of Markov’s inequality and Borel-Cantelli thatlim sup n →∞ n log H n ≤ log q − almost surely. This immediately implies the claim. (cid:3) Acknowledgments.
This work was carried out while OA was visiting and TH was an intern atMicrosoft Research, Redmond. OA is supported in part by NSERC. TH was also supported by aMicrosoft Research PhD Fellowship.
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E-mail address : [email protected] Alexander E. Holroyd
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