The random interchange process on the hypercube
aa r X i v : . [ m a t h . P R ] F e b THE RANDOM INTERCHANGE PROCESS ON THE HYPERCUBE
ROMAN KOTECK ´Y, PIOTR MI LO´S, AND DANIEL UELTSCHI
Abstract.
We prove the occurrence of a phase transition accompanied by the emergenceof cycles of diverging lengths in the random interchange process on the hypercube. Introduction
The interchange process is defined on a finite graph. With any edge is associated thetransposition of its endvertices. The outcomes of the interchange process consist of sequencesof random transpositions and the main questions of interest deal with the cycle structure ofthe random permutation that is obtained as the composition of these transpositions. As thenumber of random transpositions increases, a phase transition may occur that is indicated bythe emergence of cycles of diverging lengths involving a positive density of vertices.The most relevant graphs are regular graphs with an underlying “geometric structure” like afinite cubic box in Z d with edges between nearest neighbours. But the problem of proving theemergence of long cycles is out of reach for now and recent studies have been devoted to simplergraphs such as trees [4, 12] and complete graphs [13, 5, 6]. (Note also the intriguing identitiesof Alon and Kozma based on the group structure of permutations [3].) The motivation for thepresent article is to move away from the complete graph towards Z d . We consider the hypercube { , } n in the large n limit and establish the occurrence of a phase transition demonstratedby the emergence of cycles larger than 2 ( − ε ) n . Our proof combines the recent method ofBerestycki [5], which was used for the complete graph but is valid more generally, with anestimate of the rate of splittings that involves the isoperimetric inequality for hypercubes.Besides its interest in probability theory, the random interchange process appears in studiesof quantum spin systems [14], see also [11] for a review. Cycle lengths and cycle correlationsgive information on the magnetic properties of the spin systems. The setting is a bit different,though. First, the number of transpositions is not a fixed parameter, but a Poisson randomvariable. Second, there is an additional weight of the form θ with θ = 2 in the caseof the spin quantum Heisenberg model. The first feature is not a serious obstacle, butthe second feature turns out to be delicate. Notice that Bj¨ornberg recently obtained resultsabout the occurrence of macroscopic cycles on the complete graph in the case θ > θ = 3 , , ... , andwith a more general class of random loop models, can be found in [1, 15]. We expect that ourhypercube results can be extended to these situations as well, but it may turn out not to beentirely straightforward. 2. Setting and results
Let G n = ( Q n , E n ) be a graph whose N = 2 n vertices form a hypercube Q n = { , } n with edges joining nearest-neighbours—pairs of vertices that differ in exactly one coordinate, E n = {{ x, y } : x, y ∈ Q n , | x − y | = 1 } , | E n | = Nn . Mathematics Subject Classification.
Key words and phrases.
Random interchange, random stirring, long cycles, quantum Heisenberg model.c (cid:13)
Let Ω n be the set of infinite sequences of edges in E n . For t ∈ N by F n,t we denote the σ -algebra generated by the first t elements of the sequence. Further, for t ∈ N we use Ω n,t todenote the set of sequences of t edges e = ( e , . . . , e t ), where e s ∈ E n for all s = 1 , . . . , t . The σ -algebra F n,t will be identified with the total σ -algebra over Ω n,t . For an event A ∈ F n,t weset P n ( A ) = | A | (cid:0) Nn (cid:1) t , i.e. edges are chosen independently and uniformly from E n .Using τ e to denote the transposition of the two endvertices of an edge e ∈ E n , we can viewthe sequence e ∈ Ω n,t as a series of random interchanges generating a random permutation σ t = τ e t ◦ τ e t − ◦ · · · ◦ τ e on Q n . For any ℓ ∈ N , let V t ( ℓ ) be the random set of vertices thatbelong to permutation cycles of lengths greater than ℓ in σ t .We start with the straightforward observation that only small cycles occur in σ t when t issmall. It is based on the fact that the random interchange model possesses a natural percolationstructure when viewing any edge contained in e as opened. The probability that a particularedge remains closed by the time t is (cid:0) − Nn/ (cid:1) t . Since the set of vertices of any cycle mustbe contained in a single percolation cluster, only small cycles occur when percolation clustersare small. Theorem 2.1.
Let c < / and ǫ > . Then there exists n such that P n ( | V t ( κn ) | = 0) > − ǫκ − / for all t ≤ cN , all κ ≥ − c ) , and all n > n .Proof. In view of the above mentioned percolation interpretation of the random interchangemodel, the claim follows from the fact that the percolation model on the hypercube graph Q n is subcritical for p = 2 c/n with c < / n (see[2]). The value p = 2 c/n corresponds to t = cN implying that the probability of any particularedge to be open is 1 − (cid:0) − Nn (cid:1) cN ∼ cn . The claim of the theorem follows from [9, Theorem9]. In particular, the last displayed inequality in its proof can be reinterpreted as a claim that E n ( | V t ( κn ) | ) ≤ ǫ ( n ) κ − / (2.1) with ǫ ( n ) → n → ∞ whenever κ > − c ) . (cid:3) Our main result addresses the emergence of long cycles for large times, t > N/
2. We expectthat cycles of order N occur for all large times; here we prove a weaker claim: cycles largerthan N − ε occur for a “majority of large times”. Theorem 2.2.
Let c > and let (∆ n ) be a sequence of positive numbers such that ∆ n n/ log n →∞ as n → ∞ . Then there exist η ( c ) > and n such that for all n > n , all T > cN , and all a > , we have n T ⌊ (1+∆ n ) T ⌋ X t = T +1 E n (cid:16) | V t ( N a ) | N (cid:17) ≥ η ( c ) − a. For c > , we can take η ( c ) = (1 − c ) . Let us observe that the highest achievable value of the exponent a is just below 1 /
2; this canbe accomplished only with c becoming large. But we expect that the size of the long cycles isof order N . In fact, one can formulate a precise conjecture, namely that the joint distributionof the lengths of long cycles is Poisson-Dirichlet. This was proved in the complete graph [13],and advocated in Z d with d ≥ HE RANDOM INTERCHANGE PROCESS ON THE HYPERCUBE 3
The proof of Theorem 2.2 can be found in Section 3.4; it is based on a series of lemmasobtained in the next section.We can choose ∆ n ≡ ∆ >
0, rather than a sequence that tends to 0. In this case, Theorem2.2 takes a simpler form, which perhaps expresses the statement ‘long cycles are likely’ moredirectly.
Corollary 2.3.
Let a ∈ (0 , / , ∆ > , and ǫ ∈ (0 , − a ) . Then there exists c > and ǫ > such that for n large enough we have T ⌊ (1+∆) T ⌋ X t = T +1 P n (cid:16) | V t ( N a ) | N ≥ ǫ (cid:17) ≥ ǫ for all T > cN .Proof.
This follows from Theorem 2.2 and Markov’s inequality. Namely, P n (cid:16) | V t ( N a ) | N ≥ ǫ (cid:17) = 1 − P n (cid:16) − | V t ( N a ) | N ≥ − ǫ (cid:17) ≥ − − ǫ E n (cid:16) − | V t ( N a ) | N (cid:17) ≥ − − η ( c ) + a − ǫ . (2.2) This is positive for ǫ < η ( c ) − a . (cid:3) Occurrence of long cycles
Number of cycles vs number of clusters.
Cycle structure and percolation propertiesare intimately related, and we will rely on Berestycki’s key observation that the number ofcycles remains close to the number of clusters [5]. Let N t denote the random variable for thenumber of cycles of the random permutation σ t at time t , and e N t the number of clusters ofthe underlying percolation model. Notice that N t ≥ e N t .Let us consider the possible outcomes when a new random transposition arrives at time t .There are three possibilities; the endpoints of a new edge e t are either both in the same cycleof σ t − (and thus also in the same cluster), or in the same cluster but in different cycles, or indifferent clusters. Correspondingly, we are distinguishing three events: • S t , a splitting of a cycle where N t = N t − + 1 and e N t = e N t − . Indeed, a splitting ofany cycle necessarily occurs within the same percolation cluster. • M t , a merging of two cycles within the same cluster: N t = N t − − e N t = e N t − . • f M t , a merging of two cycles in distinct clusters: N t = N t − − e N t = e N t − − I t = S t ∪ M t be the event where the endpoints of the edge e t belong to the samecluster. Notice that f M t = I c t . Obviously, the three events above are mutually disjoint andcover all outcomes, Ω n,t = S t ∪ M t ∪ f M t . (3.1) Notice that N t − e N t = t X i =1 ( S i − M i ) . (3.2) A key in the proof of Theorem 2.2 is the isoperimetric inequality of the hypercube Q n . Namely,for any set A ⊂ Q n , the number | E ( A ) | of edges of G n whose both end-vertices are in A is | E ( A ) | ≤ | A | log | A | . (3.3) ROMAN KOTECK´Y, PIOTR MI LO´S, AND DANIEL UELTSCHI
Here (and elsewere in this paper) log is always meant as the logarithm of base 2. See [10] forthe proof of the bound in this form. It implies a lower bound on the number | E ( A | A c ) | ofedges connecting A with its complement A c = Q n \ A , namely | E ( A | A c ) | ≥ | A | ( n − log | A | ) . (3.4) We are not referring to this inequality in this article, but we found it useful in discussions.Theorem 2.2 would follow from the following lemma once its assumption is proven.
Lemma 3.1.
Assume that P n ( S t ) ≥ λ with λ ∈ (0 , . Then E n (cid:16) | V t ( N a ) | N (cid:17) ≥ λ − a − a for any a ∈ (0 , λ ) .Proof. Let C t − denote the set of cycles at time t −
1. Since the total number of edges is
N n/ P C ∈C t − | C | log | C | edges cause a splitting, we have P n ( S t |C t − ) ≤ N n X C ∈C t − | C | log | C | . (3.5) It follows that λ ≤ P n ( S t ) = E n ( P n ( S t |C t − )) ≤ N n E n (cid:16) X C ∈C t − | C | log | C | (cid:17) = 1 N n E n (cid:16) X C ∈C t − : | C |≤ N a | C | log | C | (cid:17) + 1 N n E n (cid:16) X C ∈C t − : | C | >N a | C | log | C | (cid:17) ≤ aN E n (cid:16) X C ∈C t − : | C |≤ N a | C | (cid:17) + 1 N E n (cid:16) X C ∈C t − : | C | >N a | C | (cid:17) . (3.6) Using P C ∈C t − | C | = N and P C ∈C t − : | C | >N a | C | = | V t ( N a ) | , we get the lemma. (cid:3) What remains to be done is to establish a lower bound on the probability for an edge toconnect vertices within a cycle and thus splitting it. We will get it by combining lower boundson the probability P n ( I t ) for an edge to connect vertices within one cluster and on the rate P n ( S t ) / P n ( I t ) for those actually connecting vertices within a cycle.As it turns out, we can verify the latter lower bound only in a mean sense, averaging overan interval [ T, T + L ] where T is large. The ratio L/T can be chosen to vanish but not toofast. We will use the following corollary, whose proof is essentially a verbatim repetition of theproof above.
Corollary 3.2.
Assume that for some
T, L ∈ N , and λ ∈ (0 , , we have L T + L X t = T +1 P n ( S t ) ≥ λ. Then L T + L X t = T +1 E n (cid:16) | V t ( N a ) | N (cid:17) ≥ λ − a − a for any a ∈ (0 , λ ) . HE RANDOM INTERCHANGE PROCESS ON THE HYPERCUBE 5
Lower bound on the probability of I t . Here we show that, if the time is large enough,there is a positive probability that the vertices of a random edge belong to the same cluster.Equivalently, we need an upper bound on the probability of the event f M t = I c t that twoclusters are merging. The first lemma applies to c > ; the second lemma is restricted to c > V t denote the largest percolation cluster after t randomtranspositions. Lemma 3.3.
Assume that E n ( | ˜ V t | ) > c N for some constant c > . Then there exists c ′ > such that P n ( I t ) > c ′ c (1 − o (1)) . Proof.
It is based on [2, Remark 2], which states that there exist ε > c ′ > P n ( | W t | > N − N − ε ) = 1 − o (1) , (3.7) where W t is the set of vertices which have at least c ′ n neighbours in ˜ V t . By only consideringedges within the largest cluster, we obtain P n ( I t ) ≥ c ′ N E n ( | ˜ V t ∩ W t | ) . (3.8) Using | ˜ V t ∩ W t | ≥ | ˜ V t | − | W c t | , the lemma follows. (cid:3) Remark:
In [2] the authors use their Remark 2 as an indication that for t > N the secondlargest cluster is of size o ( N ). Actually, this has been proven in [9, Theorem 31] where itwas shown that the size of the second largest cluster is of the order at most n/ (2 c − (weadhere here to our notation with critical c = 1 / c ′ n neighbours in clusters of size at least n , with [9, Theorem 31] — which implies that thisset actually coincides with W t . (Notice that in both [2] and [9, Theorem 31], the results areactually formulated for percolation clusters on the hypercube with probability of an edge beingoccupied chosen as p = 2 c/n .)We state and prove the next lemma for the hypercube, but it actually holds for any finitegraph. Lemma 3.4.
Let t ∈ N and δ ∈ (0 , . Then P n ( f M t ) ≤ N/t + t − (1 − δ ) / + exp (cid:0) − t δ / (cid:1) . Proof.
We recall that {F t } t ≥ denotes the filtration associated with the process of adding edges(i.e. F t contains information about the first t edges), and define the random variables p t = E n ( f M t |F t − ) . (3.9) Simple but crucial observations are that for any t ∈ N , we have1 ≥ p t ≥ p t +1 and t X i =1 f M i ≤ N. (3.10) Indeed, adding an edge decreases the chance of next merging and the total number of mergingsis smaller than the size of the graph. Let us define the process { X t } t ≥ by X t = t X i =1 (cid:16) f M i − p i (cid:17) . (3.11) ROMAN KOTECK´Y, PIOTR MI LO´S, AND DANIEL UELTSCHI
One verifies that it is a martingale and | X t +1 − X t | ≤
1. By the Azuma inequality we have P n (cid:0) X t ≤ − t (1+ δ ) / (cid:1) ≤ exp (cid:0) − t δ / (2 t ) (cid:1) = exp (cid:0) − t δ / (cid:1) . Hence,1 − exp (cid:0) − t δ / (cid:1) ≤ P n (cid:16) t X i =1 (cid:0) f M i − p i (cid:1) ≥ − t (1+ δ ) / (cid:17) = P n (cid:16) t X i =1 f M i ≥ − t (1+ δ ) / + t X i =1 p i (cid:17) . (3.12) Using (3.10) we estimate further,1 − exp (cid:0) − t δ / (cid:1) ≤ P n (cid:16) t X i =1 f M i ≥ − t (1+ δ ) / + tp t (cid:17) ≤ P n (cid:0) N + t (1+ δ ) / ≥ tp t (cid:1) . (3.13) In other words P n ( p t ≥ N/t + t − (1 − δ ) / ) ≤ exp (cid:0) − t δ / (cid:1) . Finally, P n ( f M t ) = E n ( f M t )= E n ( p t ) ≤ P n (cid:0) p t ≥ N/t + t − (1 − δ ) / (cid:1) + N/t + t − (1 − δ ) / ≤ exp (cid:0) − t δ/ (cid:1) + N/t + t − (1 − δ ) / . (cid:3) Lower bound on the rate P n ( S t ) / P n ( I t ) . Let us begin with a bound on the probabilityof unfavourable splittings that result in short cycles. We define the event S ≤ kt ⊂ S t as thosesplittings that result in producing a cycle of length less than or equal to k (or in two suchcycles). Lemma 3.5.
For any n, t, k ∈ N we have P n (cid:0) S ≤ kt (cid:1) ≤ k ) n . Proof.
Given an arbitrary configuration e ∈ Ω n,t yielding a collection of cycles covering Q n ,we can find a family of sets { A i } , A i ⊂ Q n such that(a) | A i | ≤ k ;(b) P i | A i | ≤ N ;(c) S ≤ kt ≤ P i e t ∈ E ( A i ) , where e t ∈ E n is the random edge at time t .Indeed, to each cycle of length less than or equal to 2 k , we define A i to be its support. For acycle of length ℓ > k , we label its vertices consecutively by natural numbers (starting froman arbitrary one) identifying the labels j, ℓ + j, ℓ + j, . . . , j = 1 , . . . , ℓ . Denoting m = ⌊ ℓ/k ⌋ ,notice that mk ≤ ℓ < ( m + 1) k and ℓ + k < ( m + 2) k < ℓ . We cover the cycle by the followingcollection of intervals(1 , . . . , k ) , ( k + 1 , . . . , k ) , . . . , (( m − k + 1 , . . . , ( m + 1) k ) , ( mk + 1 , . . . , ℓ + k )if mk < ℓ . In the case mk = ℓ , the last interval is skipped and the collection ends with(( m − k + 1 , . . . , ( m + 1) k ). Clearly, the length of all intervals is either 2 k or, for the last one, ℓ + k − mk < k thus (a) holds. Further, (b) is implied by the fact that any site of the cycleis covered exactly twice. Moreover, any pair j < j such that j ∈ (1 , . . . ℓ ) and j − j < k isnecessarily contained in at least one of above intervals. Namely, if j ∈ ( rk + 1 , . . . , ( r + 1) k ) HE RANDOM INTERCHANGE PROCESS ON THE HYPERCUBE 7 (resp. j ∈ ( mk + 1 , . . . , ℓ ) for the last interval if mk < ℓ ), then j , j ∈ ( rk + 1 , . . . , ( r + 2) k )(resp. j , j ∈ ( mk + 1 , . . . , ℓ + k )). As a result, (c) is verified and thus we get P n ( S ≤ kt ) ≤ X i P n ( e t ∈ E ( A i )) . (3.14) The number of edges E ( A i ) induced by A i is, according to the isoperimetric inequality (3.3),bounded by | A i | log | A i | . Given that the number of all edges in Q n is Nn , we get P n ( e t ∈ E ( A i )) ≤ | A i | log | A i | Nn ≤ | A i | log(2 k ) Nn . Using also that P i | A i | ≤ N , we get the claimed bound P n ( S ≤ kt ) ≤ X i | A i | log(2 k ) N n ≤ N log(2 k ) N n = 2 log(2 k ) n . (3.15) (cid:3) The bound from Lemma 3.5 can be used to show that the number of cycles N t does notdepart too far from the number of clusters e N t . Lemma 3.6.
There exists n such that for n ≥ n and any t ∈ N we have E n ( N t − e N t ) ≤ t n ) n . Proof.
Let N ≤ n (resp. N > n ) denote the number of cycles shorter or equal to 2 n (resp. longerthan 2 n ). Obviously N t = N ≤ nt + N > nt and thus E n ( N t − e N t ) = E n ( N ≤ nt − e N t ) + E n ( N > nt ) ≤ E n (cid:16) t X i =1 S ≤ ni (cid:17) + E n ( N > nt ) . (3.16) To bound E n ( N > nt ), we simply use that N > nt ≤ N/ (2 n ) ≤ t log(2 n ) n once t > N n ) . Onthe other hand, for t ≤ N n ) we get E n ( N > nt ) ≤ /n once n is sufficiently large. Indeed,observe that N > nt ≤ | V t (2 n ) | n . Hence, we can use (2.1) with c = 1 / log(2 n ) allowing to choose κ = 2 > − c ) .The result then follows from Lemma 3.5. (cid:3) Lemma 3.7.
For any
T, L ∈ N and any n ≥ n (with n the constant from Lemma 3.6), wehave T + L X t = T +1 P n ( S t ) ≥ T + L X t = T +1 P n ( I t ) − T log(4 n ) n . Proof.
We have T + L X t = T +1 S t = T + L X t = T +1 (cid:0) S t − M t (cid:1) + T + L X t = T +1 (cid:0) S t + M t (cid:1) = (cid:0) N T + L − e N T + L − N T + e N T (cid:1) + T + L X t = T +1 I t . (3.17) The claim follows by taking expectations, using that N T + L − e N T + L ≥
0, and applyingLemma 3.6 for the expectation of N T − e N T . (cid:3) ROMAN KOTECK´Y, PIOTR MI LO´S, AND DANIEL UELTSCHI
Proof of Theorem 2.2.
We check the condition of Corollary 3.2 with λ = η ( c ), where η ( c ) = ( c ′ c if c ∈ ( , , (1 − c ) if c > . (3.18) By Lemma 3.7 and Lemmas 3.3 with 3.4, we have1 L T + L X t = T +1 P n ( S t ) ≥ L T + L X t = T +1 P n ( I t ) − TL log(4 n ) n ≥ η ( c ) − o (1) (3.19) once we choose L = ∆ n T with ∆ n n/ log n → ∞ . Theorem 2.2 now follows from Corollary 3.2,since η ( c ) − a − a > η ( c ) − a ; this actually allows to neglect the corrections o (1). (cid:3) Acknowledgments:
We are grateful to Nathana¨el Berestycki for clarifying to us that his result[5] applies to arbitrary graphs of diverging degrees. We also thank the referee for bringing ourattention to Remark 2 in [2]; this allowed us to extend Theorem 2.2 from c > c > . D.U.thanks the Newton Institute for a useful visit during the program “Random Geometry” in thespring 2015. The research of R.K. was supported by the grant GA ˇCR P201/12/2613. Theresearch of P.M was supported by the grant UMO-2012/07/B/ST1/03417. References [1] M. Aizenman, B. Nachtergaele,
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Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom, andCentre for Theoretical Study, Charles University, Prague, Czech Republic
E-mail address : [email protected]
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097Warszawa, Poland
E-mail address : [email protected] Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
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