FFIXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS
SUMIT MUKHERJEE
Abstract.
Using the recently developed notion of permutation limits this paper derives the lim-iting distribution of the number of fixed points and cycle structure for any convergent sequence ofrandom permutations, under mild regularity conditions. In particular this covers random permu-tations generated from Mallows Model with Kendall’s Tau, µ random permutations introduced in[11], as well as a class of exponential families introduced in [15]. Introduction
Study of random permutations is an area of classical interest in the intersection of Combinatoricsand Probability theory. Permutation statistics of interest is indeed a long list which includes numberof fixed points, cycle structure, length of longest increasing sub-sequence, number of descents,number of cycles, number of inversions, order of a permutation, etc. Most of this literature focuseson the case where the permutation π n is chosen uniformly at random from S n . For example it iswell known that the number of fixed points of a uniformly random permutation converges to P oi (1)in distribution. More generally, denoting the number of cycles of length l by C n ( l ), we have { C n (1) , · · · , C n ( l ) } d → { P oi (1) , P oi (1 / , · · · , P oi (1 /l ) } , where the limiting Poisson variables are mutually independent. However, not much is known in thisregard outside the realm of the uniform measure. Possibly the most widely studied non uniformprobability measure on S n is the Mallows model with Kendall’s Tau, first introduced by Mallowsin [13], which has a p.m.f. of the form M n,q ( π ) = 1 Z n,q q Inv ( π n ) . (1.1)Here Inv ( π n ) := (cid:80) ≤ i
Key words and phrases.
Combinatorial probability, Mallows model, Permutation Limit, Fixed Points, Cycle structure. a r X i v : . [ m a t h . P R ] J u l SUMIT MUKHERJEE where L ( β ) := 2 β − / sinh − ( √ e β −
1) for β >
0, and 2 | β | − / sin − ( √ − e β ) for β <
0. For thescaling when n (1 − q ( n )) → ∞ , it was shown by Bhatnagar-Peled ([5]) that1 n (cid:112) − q ( n ) LIS ( π n ) p → . The more recent work of Basu-Bhatnagar ([2]) consider the case q ( n ) = q (cid:54) = 1 is fixed, and provea weak law for LIS ( π n ) (they also derive a central limit theorem for q < q ( n ) ∈ (0 ,
1) is arbitrary.In a different direction, in [11] the authors Hoppen et al. proposed a framework where a permu-tation can be viewed as a measure. This is described below in brief:For a permutation π n ∈ S n define the measure ν π on [0 , as ν π n := 1 n n (cid:88) i =1 δ ( i/n,π n ( i ) /n ) . A sequence of permutations { π n } n ≥ with π n ∈ S n is said to converge to a measure µ , if the sequenceof probability measures ν π n converge weakly to µ . Any such limit is in M , the set of probabilitydistribution on the unit square with uniform marginals. Any µ ∈ M is called a permuton (following[11]), and it is shown in [11, Theorem 1.6] that any µ ∈ M can indeed arise as a limit of a sequenceof permutations in this manner. See [3, 11] for a more detailed introduction to permutation limits.If { π n } n ≥ is a sequence of random permutations (not necessarily in the same probability space),the sequence is said to converge to a deterministic measure µ ∈ M in probability, if the sequenceof measures ν π n converge weakly to the measure µ in probability. Equivalently, for any continuousfunction f on the unit square, one haslim n →∞ n n (cid:88) i =1 f (cid:16) in , π n ( i ) n (cid:17) p → ˆ [0 , f ( x, y ) dµ. Using the topology of permutation limits in [15] the author gave a new proof for a large deviationprinciple (originally proved in [19]), and used it to analyze a class of exponential families on thespace of permutations. The large deviation principle was re-derived in [12], where Kenyon etal. study permutation ensembles constrained to have fixed densities of finite number of patterns.It was shown by Starr in [17] that if π n is generated from a Mallows model with Kendall’s Tauwith parameter q ( n ) such that n (1 − q ( n )) → β , then the sequence of measures ν π n converge weaklyin probability to a measure µ ρβ ∈ M induced by the density ρ β ( x, y ) := ( β/
2) sinh( β/ e β/ cosh( β ( x − y ) / − e − β/ cosh( β ( x + y − / , (1.2)which is the Frank’s Copula (see [16]). Since π n converges weakly to the measure µ ρβ , in anattempt to understand the marginal distribution of π n ( i ) one might conjecture that P n ( π n ( i ) = j ) ≈ n ρ β ( i/n, j/n ). We will show that this is indeed true, under certain regularity of the law ofthe random permutations. We start by introducing some notations. Definition 1.1.
For l ∈ [ n ] := { , , · · · , n } let S ( n, l ) := { p := ( p , p , · · · , p l ) ∈ [ n ] l : p a (cid:54) = p b for all a (cid:54) = b, a, b ∈ [ l ] } . IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 3
Then we have |S ( n, l ) | = (cid:0) nl (cid:1) l !. For p , q ∈ S ( n, l ) let || p − q || ∞ := max a ∈ [ l ] | p a − q a | . Also for p ∈ S ( n, l ) let π n ( p ) denote the vector ( π n ( p ) , · · · , π n ( p k )).For every n ≥ π n be a random permutation on S n with law P n . In [3, Def 6.2] the authorsdefine a notion of equi-continuity of random permutations, which they show is implied by thecondition lim δ → lim n →∞ sup p , q , r ∈S ( n,l ): || p − r || ∞ ≤ nδ (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) P n ( π n ( r ) = q ) − (cid:12)(cid:12)(cid:12) = 0 (1.3)(see [3, Prop 6.2]). In particular for l = 1 condition (1.3) in spirit demands that the function P n ( π n ( p ) = q ) is equi-continuous in p . In this paper we will need an extra notion of equi-continuitywhich demands that the function P n ( π n ( p ) = q ) is jointly equi-continuous in p, q . This is statedbelow: Definition 1.2.
A sequence of random permutations π n is said to be equi-continuous in bothco-ordinates if lim δ → lim n →∞ sup p , q , r , s ∈S ( n,l ): || p − r || ∞ ≤ nδ, || q − s ||≤ nδ (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) P n ( π n ( r ) = s ) − (cid:12)(cid:12)(cid:12) = 0 . (1.4) Definition 1.3.
Let C denote the set of all strictly positive continuous functions ρ on [0 , withuniform marginals, i.e. ˆ ρ ( x, y ) dx = ˆ ρ ( x, y ) dy = 1 . Denote by µ ρ ∈ M the measure induced by ρ .Our first theorem now proves an estimate of P n ( π n ( p ) = q ) for vectors p , q if π n is equi-continuous in both co-ordinates, and converges in the sense of permutation limits to µ ρ . Theorem 1.1.
Suppose { π n } n ≥ is a sequence of random permutations with π n ∈ S n , such thatthe sequence is equi-continuous in both co-ordinates, i.e. it satisfies (1.4) . If { π n } n ≥ converges to µ ρ for some ρ ∈ C , we have lim n →∞ sup p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) n l P n ( π n ( p ) = q ) (cid:81) la =1 ρ (cid:16) p a n , q a n (cid:17) − (cid:12)(cid:12)(cid:12) = 0 . (1.5)As an immediate corollary of Theorem 1.1 we obtain limiting distribution of the vector π n ( p ).A more general version of this corollary was already derived in [3, Proposition 6.1]. Corollary 1.2.
Suppose p n ∈ S ( n, l ) is such that lim n →∞ n p n = x ∈ [0 , l . If { π n } n ≥ is a sequence of random permutations with π n ∈ S n which satisfies (1.5) for some ρ ∈ C ,then n π n ( p n ) d → { Y ( x ) , · · · , Y ( x l ) } , where { Y ( x a ) } la =1 are mutually independent with Y ( x a ) having the density ρ ( x a , . ) . Having proved Theorem 1.1 we now turn our focus on the number of fixed points, or moregenerally the statistic N n ( π n , σ n ) := n (cid:88) i =1 { π n ( i ) = σ n ( i ) } SUMIT MUKHERJEE for any σ n ∈ S n , which denotes the number of overlaps between π n and σ n . In this notation thenumber of fixed points of π n equals N ( π n , e n ), where e n is the identity permutation in S n . By (1.5) N n ( π n , σ n ) is approximately the sum of n independent variables, and so should be approximatelydistributed as Poisson. Our next theorem confirms this conjecture, showing convergence to Poissondistribution of N n ( π n , σ n ) in distribution and in moments. Theorem 1.3.
Suppose { π n } n ≥ is a sequence of random permutations with π n ∈ S n which satisfies (1.5) for some ρ ∈ C . If σ n converges to µ , then lim n →∞ E N n ( π n , σ n ) k = E P oi ( µ [ ρ ]) k for any k ∈ N , where µ [ ρ ] := ´ [0 , ρ ( x, y ) dµ , and P oi ( λ ) is the Poisson distribution with parameter λ . Inparticular this implies N n ( π n , σ n ) d → P oi ( µ [ ρ ]) . Remark 1.1.
Setting σ n = e n it follows by Theorem 1.3 that the number of fixed points in π n hasa limiting Poisson distribution with mean ´ ρ ( x, x ) dx, provided { π n } n ≥ satisfies (1.5) for some ρ ∈ C .The random variable N n ( π n , e n ) = (cid:80) ni =1 { π n ( i ) = i } is essentially the number of cycles of length1, and a similar intuition for Poisson approximation holds for cycles of length l for any l ≥
1. Inorder to make this precise, we introduce a few more notations.
Definition 1.4.
For any l ∈ [ n ] setting U ( n, l ) := { p ∈ S ( n, l ) : p = min( p a , a ∈ [ l ]) } note that U ( n, l ) ⊂ S ( n, l ), and |S ( n, l ) | = l × |U ( n, l ) | . For p ∈ S ( n, l ) let T ( p ) ∈ S ( n, l ) denote the vector( p , p , · · · , p l , p ) . As an example if l = 3 and n = 6 then the vector p = (2 , , ∈ U ( n, l ), as 2 = min(2 , , T ( p ) = (5 , , ∈ S ( n, l ) but does not belong to U ( n, l ), as 5 (cid:54) = min(2 , , T isthe shift operator which shifts every co-ordinate by 1.For any l ≥ C n ( l ) := (cid:88) p ∈U ( n,l ) { π n ( p ) = T ( p ) } = 1 l (cid:88) p ∈S ( n,l ) { π n ( p ) = T ( p ) } . Then C n ( l ) is the number of cycles of length l , where the factor l in the second definition accountsfor the fact that every cycle is counted l times in the second sum. In particular we have C n (1) = N n ( π n , e n ) to be the number of fixed points. Also let c ρ ( l ) := 1 l ˆ [0 , l ρ ( x , x ) · · · , ρ ( x l , x ) dx · · · dx l . The following theorem derives the limiting distribution for C n ( l ) under condition (1.5). Theorem 1.4.
Suppose { π n } n ≥ is a sequence of random permutations with π n ∈ S/ n whichsatisfies (1.5) for some ρ ∈ C . Then for any { k , · · · , k l } ∈ N l we have lim n →∞ E l (cid:89) a =1 C n ( a ) k a = l (cid:89) a =1 E P oi ( c ρ ( a )) k a . In particular this implies (cid:110) C n (1) , · · · , C n ( l ) } d → (cid:110) P oi ( c ρ (1)) , · · · , P oi ( c ρ ( l )) (cid:111) , where { P oi ( c ρ ( i )) } li =1 are mutually independent. IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 5
Remark 1.2.
Thus the number of cycles of length l has a limiting Poisson distribution withparameter c ρ ( l ), whenever the sequence of permutations π n satisfies (1.5) for some ρ ∈ C . Inparticular if π n is uniformly random then (1.5) holds for the function ρ ≡
1, in which case c ρ ( l ) = l for all l ≥
1. In this case we get back the classical result that the number of cycles of length l isasymptotically P oi (1 /l ), and the random variables { C n (1) , · · · , C n ( l ) } are mutually asymptoticallyindependent for any l ∈ N .1.1. Applications.
As applications of Theorem 1.3 and Theorem 1.4, we will now derive the limitdistributions of the number of fixed points and cycle structures for three classes of non uniformdistributions on S n .(i) The first result in this direction is the next corollary, which deals with the Mallows modelwith Kendall’s Tau. Corollary 1.5.
Suppose π n is a random permutation on S n generated from the Mallowsmodel with Kendall’s Tau defined in (1.1) , such that n (1 − q ( n )) → β ∈ ( −∞ , ∞ ) . In thiscase the following conclusions hold with ρ β as defined in (1.2) .(a) If { σ n } n ≥ is a sequence of non random permutations with σ n ∈ S n converging to µ ,then N n ( π n , σ n ) converges to P oi ( µ [ ρ β ]) in distribution and in moments.(b) (cid:110) C n (1) , · · · , C n ( l ) } converges to (cid:110) P oi ( c ρ β (1)) , · · · , P oi ( c ρ β ( l )) (cid:111) in distribution and inmoments, where { P oi ( c ρ β ( i )) } li =1 are mutually independent. As an illustration of the Poisson approximation, in figure 1 we compare the histogramof the number of fixed points in a permutation of size n = 100 with the limiting Poissonprediction. We used 10000 independent observations from the Mallows model with Kendall’sTau with parameter q ( n ) = e − /n . From the picture it seems that the Poisson predictionis fairly accurate for n = 100. Since q ( n ) < f on the unit square, let Q n,θ be a one parameter exponentialfamily with sufficient statistic n (cid:88) i =1 f (cid:16) in , π n ( i ) n (cid:17) = nν π n [ f ] . More precisely, the p.m.f. is given by Q n,θ,f ( π ) = e nθν πn [ f ] − Z n ( f,θ ) , (1.6)where Z n ( f, θ ) is the log normalizing constant of the model. In particular the permutationmodel obtained by the following two specific choices have been studied in the Statisticsliterature:(a) f ( x, y ) = | x − y | , which gives the statistic (cid:80) ni =1 | π ( i ) − i | known as the Spearman’sFootrule.(b) f ( x, y ) = ( x − y ) , which gives the statistic (cid:80) ni =1 ( π ( i ) − i ) known as Spearman’s rankcorrelation Statistic.See [8, Chapter 5,6] for more on these and other non uniform permutation models con-sidered in the Statistics literature. The convergence of a sequence of random permutations π n generated from Q n,f,θ of (1.6) was shown in [15, Theorem 1.4]. Building on this result, SUMIT MUKHERJEE
Figure 1.
Bar plot of empirical distribution (from 10000 observations) of number of fixedpoints in a permutation of size n = 100 from the Mallows model with Kendall’s Tau withparameter q n = e − /n in green, compared to the Poisson prediction in yellow. the next corollary derives the limiting distributions of the number of fixed points and cyclestructure for a permutation π n generated from this model. Corollary 1.6.
Suppose π n is a random permutation on S n generated from the model Q n,f,θ defined in (1.6) for some function f which is continuous on the unit square. In this casethe following conclusions hold:(a) The sequence { π n } n ≥ converges weakly to a non random measure µ f,θ ∈ M with acontinuous density g f,θ ( ., . ) .(b) If { σ n } n ≥ is a sequence of non random permutations with σ n ∈ S n converging to µ ,then N n ( π n , σ n ) converges to P oi ( µ [ g f,θ ]) in distribution and in moments.(c) (cid:110) C n (1) , · · · , C n ( l ) } converges to (cid:110) P oi ( c g f,θ (1)) , · · · , P oi ( c g f,θ ( l )) (cid:111) in distribution andin moments, where { P oi ( c g f,θ ( i )) } li =1 are mutually independent. (iii) The final class of permutation models that we consider is a non parametric model with ameasure as the parameter, as opposed to the previous two models which are one parametermodels. This class of models will be referred to as µ random permutations, and was firstintroduced in [11].Given any µ ∈ M let ( X , Y ) , · · · , ( X n , Y n ) be i.i.d. random vectors with law µ . Definea permutation π µn ∈ S n as follows:If there exists l ∈ [ n ] such that X l = X ( i ) , Y l = Y ( j ) , then set π µn ( i ) = j . To visualizethis definition differently, let σ x and σ y be the permutations of order n such that x σ x (1) Suppose π n is a µ ρ random permutation in S n for some ρ ∈ C . In this casethe following conclusions hold:(a) If { σ n } n ≥ is a sequence of non random permutations with σ n ∈ S n which converges to µ , then N n ( π n , σ n ) converges to P oi ( µ [ ρ ]) in distribution and in moments.(b) (cid:110) C n (1) , · · · , C n ( l ) } converges to (cid:110) P oi ( c ρ (1)) , · · · , P oi ( c ρ ( l )) (cid:111) in distribution and inmoments, where { P oi ( c ρ ( i )) } li =1 are mutually independent. Even though the weak convergence of the random permutation sequence is the main ingredientin all the above results, the equi-continuity in both co-ordinates is not just a technical requirement.The following example shows that the conclusions of Theorems 1.1 and 1.3 might not hold if theequi-continuity condition fails. Proposition 1.8. Let R n,θ be a probability distributon on S n with the p.m.f. R n,θ ( π n ) = e θN n ( π n ,e n ) − Z n ( θ ) where e n is the identity permutation, and N n ( π n , e n ) is the number of fixed points in π n . Then forevery θ (cid:54) = 0 the following conclusions hold:(a) The random variable N n ( π n , e n ) converges to a Poisson random variable with mean e θ indistribution and in moments.(b) π n converges weakly to u , the uniform distribution on [0 , which is free of θ .(c) R n,θ ( π n (1) = 1 , π n (2) = 2) R n,θ ( π n (1) = 2 , π n (2) = 1) = e θ (cid:54) = 1 . Remark 1.3. Thus even though the sequence of random permutations under R n,θ converge toLebesgue measure (which is free of θ and has a continuous density), the number of fixed pointshas a limiting Poisson distribution which depends on θ . This is the case as equi-continuity in bothcoordinates does not hold here, as demonstrated by part (c) of the proposition.1.2. Scope of future research. For the Mallows model with Kendall’s Tau, the results of thispaper only apply for the case n (1 − q ( n )) = O (1). If n (1 − q ( n )) → ∞ , one should expect thenumber of fixed points to go to + ∞ , and computing the weak limits/limiting distribution aftercentering/scaling in this case remain open. In another vein, one might expect that convergencein the sense of permutations along with “mild” regularity conditions imply the weak convergenceof LIS , as worked out for the Mallows model with Kendall’s Tau in [14]. Finally, computing thelimiting density for the model defined in (1.6) might help give a more explicit description for theparameters of the limiting distributions of Corollary 1.6, as well as give non trivial copulas (bivariatedistributions with uniform marginals) which constitute a subject area of its own in Finance. SUMIT MUKHERJEE Outline of the paper. Section 2 gives the proof of Theorem 1.1, Corollary 1.2, and Theorems1.3 and 1.4. Section 3 concludes the paper by proving Corollaries 1.5-1.7, and Proposition 1.8.2. Proofs of main results Proof of Theorem 1.1 and Corollary 1.2. Proof of Theorem 1.1. For k ∈ N setting (cid:15) n ( k ) := sup p , q , r , s ∈S ( n,l ): || p − r || ∞ ≤ n/k (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) P n ( π n ( r ) = s ) − (cid:12)(cid:12)(cid:12) , condition (1.4) can be stated as lim k →∞ lim n →∞ (cid:15) n ( k ) = 0 . (2.1)Fix k ∈ N and partition (0 , 1] as ∪ ka =1 I i with I a := (cid:16) i − k , ik (cid:105) . Setting A kn := l (cid:89) a =1 I (cid:100) kp a /n (cid:101) , B kn := l (cid:89) a =1 I (cid:100) kq a /n (cid:101) note that n p ∈ A kn , n q ∈ B kn . Now for any r , s ∈ S ( n, l ) such that n r ∈ A kn , n s ∈ B kn we have (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) P n ( π n ( r ) = s ) − (cid:12)(cid:12)(cid:12) ≤ (cid:15) n ( k ) , which on summing over r ∈ A kn , s ∈ B kn and noting that the number of terms summed is at least( n − l k − l gives P n ( π n ( p ) = q ) ≤ (1 + (cid:15) n ( k )) k l ( n − l (cid:88) r ∈ A kn , s ∈ B kn P n ( π n ( r ) = s )=(1 + (cid:15) n ( k )) k l n l ( n − l E ν π ( l ) n [ A kn × B kn ]=(1 + (cid:15) n ( k )) k l n l ( n − l E l (cid:89) a =1 ν π n [ I (cid:98) kp a /n (cid:99) × I (cid:98) kq a /n (cid:99) ] , where ν lπ n denotes the l fold product measure of ν π n . Using the fact that ρ is the density for µ ρ this readily givessup p , q ∈S ( n,l ) n l P n ( π n ( p ) = q ) (cid:81) la =1 ρ ( p a n , q a n ) ≤ (1 + (cid:15) n ( k )) × n l ( n − l (2.2) × E sup i , j ∈ [ k ] l (cid:81) la =1 ν π n [ I i a × I j a ] (cid:81) li =1 µ ρ [ I i a × I j a ] (2.3) × sup x , y , z , w ∈ [0 , l : || x − z || ∞ ≤ /k, || y − w || ∞ ≤ /k (cid:81) la =1 ρ ( x a , y a ) (cid:81) la =1 ρ ( z a , w a ) . (2.4)The term in the r.h.s. of (2.2) converges to 1 on letting n → ∞ followed by k → ∞ , using (2.1).Since ν π n converges to µ ρ , by [11, Theorem 5.2] we havemax a ∈ [ l ] max i a ∈ [ k ] (cid:12)(cid:12)(cid:12) ν π n [ I i a × I j a ] − µ ρ [ I i a × I j a ] (cid:12)(cid:12)(cid:12) p → , IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 9 which along with the observation that µ ρ [ I i a × I j a ] is uniformly bounded away from 0 givesmax a ∈ [ l ] max i a ∈ [ k ] (cid:81) la =1 ν π n [ I i a × I j a ] (cid:81) la =1 µ ρ [ I i a × I j a ] p → n → ∞ , for k fixed. An application of Dominated Convergence theorem implies that the term in(2.3) converges to 1 as well. Finally (2.4) is free of n , and converges to 1 as k → ∞ by continuityof ρ . Combining this giveslim sup k →∞ lim sup n →∞ sup p , q , r ∈S ( n,l ): || p − r || ∞ ≤ n/k n l P n ( π n ( p ) = q ) (cid:81) la =1 ρ ( p a n , q a n ) ≤ , thus giving the upper bound in (1.5). A similar proof gives the lower bound, thus completing theproof of the theorem. (cid:3) We now introduce some auxiliary variables, to be used in the proofs of Corollary 1.2, and Theo-rems 1.3 and 1.4. Definition 2.1. For every n ≥ { Z n (1) , · · · , Z n ( n ) } be mutually independent random variablessupported on [ n ] such that the marginal laws are given by Q n ( Z n ( p ) = q ) = ρ ( p/n, q/n ) (cid:80) ns =1 ρ ( p/n, q/n )for some ρ ∈ C . Also set M n ( σ n ) := n (cid:88) p =1 { Z n ( p ) = σ n ( p ) } , and for l ≥ D n ( l ) := (cid:88) p ∈U ( n,l ) { Z n ( p ) = T ( p ) } . Proof of Corollary 1.2. With Z n as constructed in definition 2.1 we have (cid:88) q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p n ) = q ) − Q n ( Z n ( p n ) = q ) (cid:12)(cid:12)(cid:12) ≤ max p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p n ) = q ) Q n ( Z ( p n ) = q ) − (cid:12)(cid:12)(cid:12) (cid:88) q ∈S ( n,l ) Q n ( Z n ( p n ) = q )= max p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p n ) = q ) Q n ( Z n ( p n ) = q ) − (cid:12)(cid:12)(cid:12) , which goes to 0 by (1.5). This implies that the laws of π n ( p n ) and Z n ( p n ) are close in totalvariation. Since the desired conclusion can be verified easily for Z n ( p n ), the proof is complete. (cid:3) Proofs of Theorem 1.3 and 1.4. We will use Stein’s method based on dependency graphsto prove Poisson limit theorems, as explained below:Let { X α } α ∈ I be a finite set of Bernoulli random variables. A dependency graph for { X α } α ∈ I isa graph with node set I and edge set E , such that if I , I are disjoint subsets of I with no edgesconnecting them, then { X α } α ∈ I and { X β } β ∈ I are independent. Let N α be the neighborhoodof vertex α , i.e. N ( α ) := { β ∈ I : ( α, β ) ∈ E } ∪ { α } . Then one has the following Poissonapproximation result, first proved in [1]. Theorem 2.1. [7, Theorem 15] Let { X α } α ∈ I be a finite set of Bernoulli random variables withdependency graph ( I, E ) . Then setting λ := (cid:80) α ∈ I p α , W := (cid:80) α ∈ I X α we have ||L ( W ) − L ( P oi ( λ )) || T V ≤ (cid:88) α ∈ I (cid:88) β ∈ N ( α ) / { α } E X α X β + (cid:88) α ∈ I (cid:88) β ∈ N ( α ) E X α E X β . The following lemma uses Theorem 2.1 to prove two Poisson limits which will be used in theproofs of Theorems 1.3 and 1.4. Lemma 2.1. Let M n ( σ n ) and D n ( l ) be as in definition 2.1.(a) If σ n converges to µ ∈ M in the sense of permutation limits, then we have M n ( σ n ) d → P µ [ ρ ] ,and lim n →∞ E M n ( σ n ) k = E P oi ( µ [ ρ ]) k , for all k ∈ N . (b) For any l ∈ N we have D n ( l ) d → P c ρ ( l ) , and lim n →∞ E D n ( l ) k = E P oi ( c ρ ( l )) k , for all k ∈ N . Proof. Setting m := inf ≤ x,y ≤ ρ ( x, y ), M := sup ≤ x,y ≤ ρ ( x, y ) we have 0 < m ≤ M < ∞ .(a) Since the random variables X p = 1 { Z n ( p ) = σ n ( p ) } for p = 1 , , · · · , n are mutually inde-pendent, the dependency graph of { X , X , · · · , X n } is empty. It then follows by Theorem2.1 that ||L ( M n ( σ n )) − L ( P oi ( λ n )) || T V ≤ n (cid:88) p =1 (cid:104) ρ ( p/n, σ n ( p ) /n ) (cid:80) nq =1 ρ ( p/n, q/n ) (cid:105) ≤ n × M m where λ n = n (cid:88) p =1 ρ ( p/n, σ n ( p ) /n ) (cid:80) nq =1 ρ ( p/n, q/n ) n →∞ → ˆ [0 , ρ ( x, y ) dµ = µ [ ρ ] , and so M n ( σ n ) converges to P oi ( µ [ ρ ]) in distribution. To conclude convergence in momentsit suffices to show that lim sup n →∞ E M n ( σ n ) k < ∞ for every k ∈ N . To see this, set (cid:101) S ( n, l ) := { p ∈ S ( n, l ) : p < p < · · · < p l } denote the set of all n tuples in increasing order, and note that E M n ( σ n ) k = (cid:88) p ∈ [ n ] k Q n ( Z n ( p ) = σ n ( p )) ≤ k (cid:88) l =1 k l (cid:88) p ∈ (cid:101) S ( n,l ) Q n ( Z n ( p ) = σ n ( p )) . Here the factor k l in the r.h.s. above accounts for the fact that a specific term { Z n ( p ) = σ n ( p ) } with p ∈ (cid:101) S ( n, l ) can arise from at most k l terms in [ n ] k . Since | (cid:101) S ( n, l ) | = (cid:0) nl (cid:1) , wecan bound the r.h.s. above by k (cid:88) l =1 k l (cid:88) p ∈ (cid:101) S ( n,l ) l (cid:89) a =1 ρ ( p a /n, σ n ( p a ) /n ) (cid:80) nq a =1 ρ ( p a /n, q a /n ) ≤ k (cid:88) l =1 k l l ! (cid:16) Mm (cid:17) l < ∞ . (b) The proof of part (b) is similar to the proof of part (a). For p ∈ U ( n, l ) setting X p =1 { Z n ( p ) = T ( p ) } note that X p is independent of X q whenever the indices p and q have nooverlap. Thus the dependency graph of the random variables { X p , p ∈ U n,l } has maximum IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 11 degree at most (cid:0) n − l − (cid:1) l !. Also for any p , q which overlap we have E X p X q = 0 unless p = q .Thus an application of Theorem 2.1 gives ||L ( D n ( l )) − L ( P λ n ) || ≤ (cid:18) nl (cid:19) ( l − × (cid:18) n − l − (cid:19) l ! × M l n l m l ≤ n × M l m l , with λ n = 1 l (cid:88) p ∈S n,l ρ ( p /n, p /n ) (cid:80) nq =1 ρ ( p /n, q /n ) × · · · × ρ ( p l /n, p /n ) (cid:80) nq l =1 ρ ( p l /n, q l /n ) n →∞ → c ( l ) , and so D n ( l ) converges to P oi ( c ( l )) in distribution. Convergence in moments follows by asimilar calculation as before. (cid:3) Proof of Theorem 1.3. Let { Z n (1) , · · · , Z n ( n ) } and M n ( σ n ) be as defined in 2.1. Then using part(a) of Lemma 2.1 and the fact that the Poisson distribution is characterized by its moments, itsuffices to show that for every k ∈ N we havelim n →∞ | E N n ( π n , σ n ) k − E M n ( σ n ) k | = 0 . To this effect setting Z n ( p ) = ( Z n ( p ) , · · · , Z n ( p k )) for p ∈ [ n ] k we have | E N n ( π n , σ n ) k − E M n ( σ n ) k | ≤ (cid:88) p ∈ [ n ] k (cid:12)(cid:12)(cid:12)(cid:110) P n (cid:16) π n ( p ) = σ n ( p ) (cid:17) − Q n (cid:16) Z n ( p ) = σ n ( p ) (cid:17)(cid:111)(cid:12)(cid:12)(cid:12) First note that the events { π n ( p ) = σ n ( p ) } and { Z n ( p ) = σ n ( p ) } have positive probability for all p ∈ [ n ] k , and so for any p ∈ [ n ] k setting L = L ( p ) denote the number of distinct indices gives thebound (cid:12)(cid:12)(cid:12)(cid:110) P n (cid:16) π n ( p ) = σ n ( p ) (cid:17) − Q n (cid:16) Z n ( p ) = σ n ( p ) (cid:17)(cid:111)(cid:12)(cid:12)(cid:12) ≤ max p , q ∈S ( n,L ) (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) Q n ( Z n ( p ) = q ) − (cid:12)(cid:12)(cid:12) Q n (cid:16) Z n ( p ) = σ n ( p ) (cid:17) . Since L ( p ) ≤ k , taking a maximum over L and summing over p ∈ [ n ] k gives the bound | E N n ( π n , σ n ) k − E M n ( σ n ) k | ≤ (cid:26) max l ∈ [ k ] max p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) Q n ( Z n ( p ) = q ) − (cid:12)(cid:12)(cid:12)(cid:27) E M n ( σ n ) k . (2.5)By (1.5) we have max l ∈ [ k ] max p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) Q n ( Z n ( p ) = q ) − (cid:12)(cid:12)(cid:12) → . Since Lemma 2.1 implies lim sup n →∞ E M n ( σ n ) k = E P oi ( µ [ ρ ]) k < ∞ , the r.h.s. of (2.5) converges to 0 as n → ∞ , thus completing the proof of the theorem. (cid:3) Proof of Theorem 1.4. Let { Z n (1) , · · · , Z n ( n ) } and { D n ( a ) , ≤ a ≤ l } be as defined in 2.1. Thenby part (b) of Lemma 2.1, for any finite collection of non negative integers k , k , · · · , k l we havelim n →∞ l (cid:89) a =1 E D n ( a ) k a = l (cid:89) a =1 E P oi ( c ρ ( a )) k a . Thus to complete the proof it suffices to show the following:lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E l (cid:89) a =1 D n ( a ) k a − l (cid:89) a =1 E D n ( a ) k a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (2.6)lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E l (cid:89) a =1 C n ( a ) k a − E l (cid:89) a =1 D n ( a ) k a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (2.7)For showing (2.6) we have | E l (cid:89) a =1 D n ( a ) k a − l (cid:89) a =1 E D n ( a ) k a | ≤ (cid:88) Γ (cid:12)(cid:12)(cid:12) Q n (cid:16) ∩ al =1 ∩ k a b a =1 (cid:110) Z n ( p ( a, b a )) = T ( p ( a, b a )) (cid:111)(cid:17) − l (cid:89) a =1 Q n (cid:16) ∩ k a b a =1 (cid:110) Z n ( p ( a, b a )) = T ( p ( a, b a )) (cid:111)(cid:17)(cid:12)(cid:12)(cid:12) , (2.8)where Γ := (cid:110) p ( a, b a ) ∈ U ( n, a ) , b a = 1 , , · · · , k a , a = 1 , , · · · , l (cid:111) . Proceeding to analyze a generic term in the r.h.s. of (2.8), fix p ( a, b a ) ∈ U ( n, a ) , ≤ b a ≤ k a , ≤ a ≤ l. Let L a = L a { p ( a, b a ) , ≤ b a ≤ k a } denote the set of distinct indices in the set { p ( a, b a ) , ≤ b a ≤ k a } . First note that if the sets L a do not overlap across a , both terms in the r.h.s. of (2.8)are the same, and so gets canceled. As an example, this happens for the choice l = 3 , k = 0 , k = 1 , k = 2 , p (2 , 1) = (1 , , p (3 , 1) = (3 , , , p (3 , 2) = (3 , , . In this case L = φ, L = { , } and L = { , , } do not overlap, and so the corresponding termsin the r.h.s. of (2.8) get cancelled.If the sets L a do overlap across a , then the first term in the r.h.s. of (2.8) is 0. In this casesetting L := (cid:80) la =1 | L a | the total contribution of the second term in the r.h.s. of (2.8) is boundedby (cid:16) Mmn (cid:17) L . Since there is a repetition among the indices, the number of distinct indices L ( D ) inthe set { p ( a, b a ) , ≤ b a ≤ k a , ≤ a ≤ l } is strictly less than L . As an example, this happens forthe choice l = 3 , k = 0 , k = 1 , k = 2 , p (2 , 1) = (1 , , p (3 , 1) = (3 , , , p (3 , 2) = (1 , , . In this case L = φ, L = { , } , L = { , , , , , } , and so the number of distinct indices L ( D ) = 7 which is less than L = | L | + | L | = 8. Setting K := (cid:80) la =1 k a , the total number ofterms with exactly L ( D ) distinct indices is at most (cid:0) nL ( D ) (cid:1) K !. Summing over the possible ranges L ( D ) ∈ [1 , L − , L ∈ [1 , K ] the total contribution of such terms is at most K (cid:88) L =1 L − (cid:88) L ( D )=1 (cid:18) nL ( D ) (cid:19) K ! (cid:16) Mmn (cid:17) L = O (cid:16) n (cid:17) , thus proving (2.6). IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 13 Proceeding to prove (2.7) we again have | E l (cid:89) a =1 C n ( a ) k a − E l (cid:89) a =1 D n ( a ) k a | ≤ (cid:88) Γ (cid:12)(cid:12)(cid:12) P n (cid:16) ∩ la =1 ∩ k a b a =1 π n ( p ( a, b a )) = T ( p ( a, b a )) (cid:17) − Q n (cid:16) ∩ la =1 ∩ k a b a =1 Z n ( p ( a, b a )) = T ( p ( a, b a )) (cid:17)(cid:12)(cid:12)(cid:12) (2.9)Proceeding to bound the r.h.s. of (2.9), note that in this case if all the indices in the set ∪ la =1 L a are not distinct (i.e. L ( D ) (cid:54) = L ), then both terms in the r.h.s. of (2.9) are 0. Even if L ( D ) = L , itis possible that both terms are 0, which happens for example for the choice l = 3 , k = 0 , k = 1 , k = 2 , p (2 , 1) = (1 , , p (3 , 1) = (3 , , , p (3 , 2) = (3 , , . In this case L = { , } , L = { , , } and so L ( D ) = L = 5. However both the terms on the r.h.s.of (2.9) have 0 probability. If either of the terms have non zero probability, then a generic termon the r.h.s. of (2.7) is of the form | P n ( π n ( p ) = q ) − Q n ( Z n ( p ) = q ) | for some p , q ∈ S ( n, l ) with l ∈ [ L ]. Noting that L ≤ K , this can be bounded bymax l ∈ [ K ] max p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) Q n ( Z n ( p ) = q ) − (cid:12)(cid:12)(cid:12) Q n (cid:16) ∩ la =1 ∩ k a b a =1 Z n ( p ( a, b a )) = T ( p ( a, b a )) (cid:17)(cid:12)(cid:12)(cid:12) . On summing over Γ using (2.9) gives | E l (cid:89) a =1 C n ( a ) k a − E l (cid:89) a =1 D n ( a ) k a | ≤ max l ∈ [ K ] max p , q ∈S ( n,l ) (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) Q n ( Z n ( p ) = q ) − (cid:12)(cid:12)(cid:12) E l (cid:89) a =1 D n ( a ) k a , from which (2.7) follows on using (1.5) along with (2.6). (cid:3) Proof of Corollaries 1.5-1.7 and Proposition 1.8 Proof of Corollary 1.5. By [17, Theorem 1] it follows that π n converges weakly in probability tothe measure µ ρ β induced by the density ρ β defined in (1.2). Given Theorems 1.1, 1.3 and 1.4, forproving both parts (a) and (b) it suffices to verify the equi-continuity condition (1.4), which isequivalent to the following two conditions:lim δ → lim n →∞ sup p , q , r ∈S ( n,l ): || p − r || ∞ ≤ nδ (cid:12)(cid:12)(cid:12) P n ( π n ( p ) = q ) P n ( π n ( r ) = q ) − (cid:12)(cid:12)(cid:12) = 0 , (3.1)lim δ → lim n →∞ sup q , r , s ∈S ( n,l ): || q − s || ∞ ≤ nδ (cid:12)(cid:12)(cid:12) P n ( π n ( r ) = q ) P n ( π n ( r ) = s ) − (cid:12)(cid:12)(cid:12) = 0 . (3.2)Recall that (3.1) was already verified in [3, Corollary 6.3+Lemma 7.1]. By repeating the argumentpresented there, we prove both (3.1) and (3.2) here for completeness. To show (3.1), fix p , q , r suchthat (cid:107) p − r (cid:107) ∞ ≤ nδ . Let Ω( p , q ) denote the set of all permutations in S n such that π n ( p ) = q ,and Ω( r , q ) be defined likewise. We will now define a bijection Φ = Φ[( p , q ); ( r , q )] from Ω( p , q ) toΩ( r , q ). For any π n ∈ Ω( p , q ) setΦ( π n )( r ) = q , Φ( π n )( p ) := π n ( r ) , Φ( π n )( i ) = π n ( i ) otherwise . It is easy to see that Φ is indeed a bijection, and M n,q ( n ) ( π n ) M n,q ( n ) (Φ( π n )) = q ( n ) Inv ( π n ) − Inv (Φ( π n )) ≤ max (cid:16) q ( n ) , q ( n ) − (cid:17) nlδ , where we use the fact that the inversion status of a pair ( i, j ) in π n is the same as its inversionstatus in Φ( π n ) unless i ∈ ∪ la =1 [ p a , r a ] and j ∈ q . Summing over π n ∈ Ω( p , q ) gives P n ( π n ( p ) = q ) P n ( π n ( r ) = q ) ≤ max (cid:16) q ( n ) , q ( n ) − (cid:17) nlδ , and since the bound in the r.h.s. above is free of p , q , r , taking a sup givessup p , q , r ∈S ( n,l ): || p − r || ∞ ≤ nδ P n ( π n ( p ) = q ) P n ( π n ( r ) = q ) ≤ max (cid:16) q ( n ) , q ( n ) − (cid:17) nlδ . On letting n → ∞ followed by δ → n (1 − q ( n )) → β ∈ ( −∞ , ∞ ), we getlim sup δ → lim sup n →∞ sup p , q , r ∈S ( n,l ): || p − r || ∞ ≤ nδ P n ( π n ( p ) = q ) P n ( π n ( r ) = q ) ≤ , thus giving the upper bound in (3.1). By symmetry we havelim inf δ → lim inf n →∞ sup p , q , r ∈S ( n,l ): || p − r || ∞ ≤ nδ P n ( π n ( p ) = q ) P n ( π n ( r ) = q ) ≥ , thus giving the lower bound, and hence proving (3.1). For proving (3.2) a similar argument works,except now we set up the bijection (cid:101) Φ n = (cid:101) Φ n [( r , q ); ( r , s )] between Ω r , q to Ω r , s by setting (cid:101) Φ( π n )( r ) = s , (cid:101) Φ( π n )( π − n s ) := q , Φ( π n )( i ) = π n ( i ) otherwise . The rest of the argument repeats itself, and we omit the details. (cid:3) Proof of Corollary 1.6. (a) It follows from [15, Theorem 1.4] that π n converges to a uniquemeasure µ f,θ weakly in probability, which is the solution of the optimization problem µ (cid:55)→ { θµ [ f ] − D ( µ || u ) } , where u is the uniform measure on the unit square, and D ( . || . ) is the Kullback Leiblerdivergence. It was further shown there that µ f,θ has a density of the form g f,θ ( x, y ) = e θf ( x,y )+ a f,θ ( x )+ b f,θ ( y ) , where a f,θ ( . ) and b f,θ ( . ) are unique almost surely. To complete theproof of part (a), it suffices to show that the function g f,θ is continuous on the unit square, orequivalently that e − a f,θ ( . ) is continuous. To this effect, using the fact that µ f,θ has uniformmarginals, we have e − a f,θ ( x ) = ˆ e θf ( x,y )+ b f,θ ( y ) dy, which readily gives ˆ e b f,θ ( y ) dy ≤ e − a f,θ ( x ) − inf x,y ∈ [0 , { θf ( x,y ) } for almost all x ∈ [0 , e b f,θ ( . ) is integrable. But then we have (cid:12)(cid:12)(cid:12) e − a f,θ ( x ) − e − a f,θ ( x ) (cid:12)(cid:12)(cid:12) ≤ sup y ∈ [0 , (cid:12)(cid:12)(cid:12) e θf ( x ,y ) − e θf ( x ,y ) (cid:12)(cid:12)(cid:12) ˆ e b f,θ ( y ) dy, from which continuity of e − a f,θ ( . ) follows from continuity of f ( ., . ). IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 15 (b),(c) As in the proof of Corollary 1.5 it suffices to verify the conditions (3.1) and (3.2). Usingthe same notations as in the proof of Corollary 1.5, we have Q n,f,θ ( π n ) Q n,f,θ (Φ( π n )) = e θ (cid:80) la =1 f ( p a /n,q a /n ) − f ( r a /n,q a /n ) , and the exponent in the r.h.s. above is bounded by | θ | sup x ,x ,y ∈ [0 , | x − x |≤ δ | f ( x , y ) − f ( x , y ) | . Since this goes to 0 as δ → 0, a similar proof as before verifies (3.1). The proof of (3.2) issimilar, and again we omit the details. (cid:3) Proof of Corollary 1.7. Since a sequence of µ ρ random permutations converge to µ ρ weakly inprobability, it suffices to verify (3.1) and (3.2).To this effect, with ( X , Y ) , · · · , ( X n , Y n ) i.i.d. ∼ µ ρ first note that marginally both ( X , · · · , X n )and ( Y , · · · , Y n ) are i.i.d. U (0 , U , · · · , U n ) and ( V , · · · , V n ) are the order statistics of( X , · · · , X n ) and ( Y , · · · , Y n ) respectively, for any δ > P n (cid:16)(cid:12)(cid:12)(cid:12) U i − in (cid:12)(cid:12)(cid:12) > δ (cid:17) = P n (cid:18) Bin (cid:16) n, in − δ (cid:17) ≥ i (cid:19) + P n (cid:18) Bin (cid:16) n, in + δ (cid:17) ≤ i (cid:19) ≤ e − δ n (3.3)by Hoeffding’s inequality. Also using (1.7), for any p , q ∈ S ( n, l ) we have P n ( π n ( p ) = q ) = n ! (cid:88) π n ∈ Ω( p , q ) ˆ u
With P n = R n, denoting the uniform measure on S n and D n denotingthe number of derangements of n , we have1 n ! e Z n ( θ ) = E P n e θN n ( π n ,e n ) = ∞ (cid:88) k =0 e θk (cid:0) nk (cid:1) D n − k n ! → exp { e θ − } , (3.7)where we use the fact that D n /n ! converges to e − .(a) For any λ > E R n,θ e λN n ( π n ,e n ) = e Z n ( θ + λ ) − Z n ( θ ) → exp { e θ ( e λ − } , and so N n ( π n , e n ) converges to P oi ( e θ ) in distribution and in moments.(b) With D ( . || . ) denoting the Kullback-Leibler divergence we have D ( R n, || R n,θ ) = log (cid:16) e Z n ( θ ) n ! (cid:17) − θ E P n N ( π n , e n ) → e θ − − θ, and so by [4, Prop 5.1] we have that the two probability distributions R n,θ and R n, = P n are mutually contiguous. Since π n converges weakly to u under P n = R n, , by contiguitythe same happens for R n,θ .(c) Let A n := { π n ∈ S n : π n (1) = 1 , π n (2) = 2 } , and B n := { π n ∈ S n : π n (1) = 2 , π n (2) = 1 } .Define a bijection ω from A n to B n by setting ω ( π n )( i ) = i for 3 ≤ i ≤ n , and note that R n,θ ( π n ) R n,θ ( ω ( π n )) = e θ , and so summing over π n ∈ A n gives R n,θ ( π n (1) = 1 , π n (2) = 2) R n,θ ( π n (1) = 2 , π n (2) = 1) = e θ (cid:54) = 1 , thus proving part (c). (cid:3) IXED POINTS AND CYCLE STRUCTURE OF RANDOM PERMUTATIONS 17 Acknowledgements The Poisson distribution for the number of fixed points in the Mallows model with Kendall’sTau was conjectured by Susan Holmes based on empirical evidence. This paper also benefited fromhelpful discussions with Shannon Starr. Suggestions from an anonymous referee greatly improvedthe presentation of the paper. References [1] Arratia, R. , Goldstein, L. , and Gordon, L. (1990). Poisson approximation and the Chen-Stein method. Statist. Sci. , 4, 403–434. With comments and a rejoinder by the authors. MR 1092983[2] Basu, R. and Bhatnagar, N. (2016). Limit Theorems for Longest Monotone Subsequences in Random MallowsPermutations. Available at http://arxiv.org/pdf/1601.02003 .[3] Bhattachara, B. and Mukherjee, S. (2015). Degree sequence of random permutation graphs. Ann. Appl.Probab. , to appear.[4] Bhattachara, B. and Mukherjee, S. (2015). Inference in Ising models. Available at http://arxiv.org/abs/1507.07055 .[5] Bhatnagar, N. and Peled, R. (2015). Lengths of monotone subsequences in a Mallows permutation. Probab.Theory Related Fields , 3–4, 719–780. MR 3334280[6] Borodin, A. , Diaconis, P. , and Fulman, J. (2010). On adding a list of numbers (and other one-dependentdeterminantal processes). Bull. Amer. Math. Soc. (N.S.) , 4, 639–670. MR 2721041[7] Chatterjee, S. , Diaconis, P. , and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 , 64–106. MR 2121796[8] Diaconis, P. (1988). Group representations in probability and statistics . Institute of Mathematical StatisticsLecture Notes—Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA. MR 0964069[9] Diaconis, P. and Ram, A. (2000) Analysis of Systematic Scan Metropolis Algorithms Using Iwahori-HeckeAlgebra Techniques. Michigan Math. J. , 1, 157–190. MR 1786485[10] Gladkich, A. and Peled, R. (2016) On the cycle structure of Mallows permutations. Available at http://arxiv.org/pdf/1601.06991 .[11] Hoppen, C. , Kohayakawa, Y. , Moreira, C. G. , R´ath, B. , and Menezes Sampaio, R. (2013). Limits ofpermutation sequences. J. Combin. Theory Ser. B , 1, 93–113. MR 2995721[12] Kenyon, R. , Kr´al, D. , Radin, C. , and Winkler, P. (2015). A variational principle for permutations. Avail-able at http://arxiv.org/pdf/1506.02340 .[13] Mallows, C. L. (1957). Non-null ranking models. I. Biometrika 44 , 114–130. MR 0087267[14] Mueller, C. and Starr, S. (2013). The length of the longest increasing subsequence of a random Mallowspermutation. J. Theoret. Probab. , 2, 514–540. MR 3055815[15] Mukherjee, S. (2016). Estimation in exponential families on permutations. Ann. Statist. , 2, 853–875.MR 3476619[16] Nelsen, R. B. (2006). An introduction to copulas , Second ed. Springer Series in Statistics. Springer, New York.MR 2197664[17] Starr, S. (2009). Thermodynamic limit for the Mallows model on S n . J. Math. Phys. , 9, 095208, 15.MR 2566888[18] Walters, M. and Starr, S. (2015). A note on mixed matrix moments for the complex Ginibre ensemble. J.Math. Phys. , 1, 013301, 20. MR 3390837[19] Trashorras, J. (2008). Large deviations for symmetrised empirical measures.