Cutoff phenomenon for the warp-transpose top with random shuffle
aa r X i v : . [ m a t h . P R ] J a n CUTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITHRANDOM SHUFFLE
SUBHAJIT GHOSH
Abstract.
Let { G n } ∞ be a sequence of non-trivial finite groups, and b G n denote the set ofall non-isomorphic irreducible representations of G n . In this paper, we study the propertiesof a random walk on the complete monomial group G n ≀ S n generated by the elements ofthe form (e , . . . , e , g ; id) and (e , . . . , e , g − , e , . . . , e , g ; ( i, n )) for g ∈ G n , ≤ i < n . Wecall this the warp-transpose top with random shuffle on G n ≀ S n . We find the spectrum ofthe transition probability matrix for this shuffle. We prove that the mixing time for thisshuffle is O (cid:0) n log n + n log( | G n | − (cid:1) . We show that this shuffle presents ℓ -pre-cutoff at n log n + n log( | G n | − ℓ -cutoff phenomenonwith cutoff time n log n + n log( | G n | −
1) if | b G n | = o ( | G n | δ n δ ) for all δ >
0. We provethat this shuffle has total variation cutoff at n log n + n log( | G n | −
1) if | G n | = o ( n δ ) for all δ > Introduction
The number of shuffles required to mix up a deck of cards has received considerableattention in the last few decades. The card shuffling problems can be described as randomwalks on the symmetric group. Random walks on finite groups generalize card shufflingmodels by replacing the symmetric group by other finite groups. The random walks on thefinite groups are well-studied topics in probability theory [19]. A random walk convergesto a unique stationary distribution subject to certain natural conditions. The mixing time(number of steps required to reach the stationary distribution up to a given tolerance) ofrandom walks has been intensively studied in recent times. To study the convergence rateof random walks, it is helpful to know the eigenvalues and eigenvectors of the transitionmatrix. Similar convergence rate questions for finite Markov chains are useful in manysubjects, including statistical physics, computer science, biology and more [17].In the eighties, Diaconis and Shahshahani introduced the use of non-commutative Fourieranalysis techniques in their work on the random transposition shuffle [7]. They proved thatthis shuffle on n distinct cards has total variation cutoff at n log n . After this landmarkwork, the theory of random walks on finite groups obtained its own independence, its ownproblems and techniques. Afterwards, some other techniques have come to deal with randomwalks on finite groups (viz. the coupling argument [1], the strong stationary time approach[2, 3]). Our model is mainly inspired by the transpose top with random shuffle on thesymmetric group S n [9]. Given a deck of n distinct cards this shuffle choose a card from the Mathematics Subject Classification.
Key words and phrases. random walk, complete monomial group, mixing time, cutoff, Young-Jucys-Murphy elements. deck uniformly at random and transpose it with the top card. This shuffle exhibits totalvariation cutoff at n log n [4, 5]. The transpose top with random shuffle has been recentlygeneralized to the cards with two orientations known as the flip-transpose top with randomshuffle on the hyperoctahedral group B n [10]. The flip-transpose top with random shuffle on B n has total variation cutoff at n log n . In this work, we generalize the flip-transpose topwith random shuffle to the complete monomial group G n ≀ S n . An extended abstract of thiswork appeared in FPSAC 2020 (online). For other works on the complete monomial groupssee [20, 8]. Before describing our random walk model, let us first recall the definition of thecomplete monomial group. Definition 1.1.
Let G be a finite group and S n be the symmetric group of permutations ofelements of the set [ n ] := { , , . . . , n } . The complete monomial group is the wreath productof G with S n , is a group denoted by G ≀ S n and can be described as follows: The elementsof G ≀ S n are ( n + 1)-tuples ( g , g , . . . , g n ; π ) where g i ∈ G and π ∈ S n . The multiplicationin G ≀ S n is given by ( g , . . . , g n ; π )( h , . . . , h n ; η ) = ( g h π − (1) , . . . , g n h π − ( n ) ; πη ). Therefore( g , . . . , g n ; π ) − = ( g − π (1) , . . . , g − π ( n ) ; π − ).Now let { G n } ∞ be a sequence of non-trivial finite groups. We consider the completemonomial groups G n := G n ≀ S n for each positive integer n . Let e be the identity of G n andid be the identity of S n . For an element π ∈ S n , let π := (e , . . . , e; π ) ∈ G n and for g ∈ G n ,let g ( i ) := (e , . . . , e , g ↑ , e , . . . , e; id) ∈ G n .i th position.Unless otherwise stated from now on, (e , . . . , e , g − , e , . . . , e , g ; ( i, n )) denotes the element of G n with g − in i th position and g in n th position, for g ∈ G n , ≤ i < n . One can checkthat ( g − ) ( i ) g ( n ) ( i, n ) is equal to (e , . . . , e , g − , e , . . . , e , g ; ( i, n )) for g ∈ G n , ≤ i < n .In this work we consider a random walk on the complete monomial group G n driven by aprobability measure P , defined as follows:(1) P ( x ) = n | G n | , if x = (e , . . . , e , g ; id) for g ∈ G n , n | G n | , if x = (e , . . . , e , g − , e , . . . , e , g ; ( i, n )) for g ∈ G n , ≤ i < n, , otherwise . We call this the warp-transpose top with random shuffle because at most times the n thcomponent is multiplied by g and the i th component is multiplied by g − simultaneously, g ∈ G n , 1 ≤ i < n . We now give a combinatorial description of this model as follows:Let A n ( G ) denote the set of all arrangements of n coloured cards in a row such that thecolours of the cards are indexed by the set G . For example, if Z denotes the additive groupof integers modulo 2, then elements of A n ( Z ) can be identified with the elements of B n (thehyperoctahedral group). For g, h ∈ G , by saying update the colour g using colour h we meanthe colour g is updated to colour g · h . Elements of G n can be identified with the elementsof A n ( G n ) as follows: The element ( g , . . . , g n ; π ) ∈ G n is identified with the arrangement in A n ( G n ) such that the label of the i th card is π ( i ), and its colour is g π ( i ) , for each i ∈ [ n ].Given an arrangement of coloured cards in A n ( G n ), the warp-transpose top with random UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 3 shuffle on G n is the following: Choose a positive integer i uniformly from [ n ]. Also choose acolour g uniformly from G n , independent of the choice of the integer i .(1) If i = n : update the colour of the n th card using colour g .(2) If i < n : first transpose the i th and n th cards. Then simultaneously update thecolour of the i th card using colour g and update the colour of the n th card usingcolour g − . 12345978612345 6 789 − − − → −−−→ −−− → − − − → −−−→ −−− → a ) ( b ) Figure 1.
Transitions for the warp-transpose top with random shuffle on Z ≀ S . Z is the additive group of integers modulo 3, consists of the coloursred, green and blue such that red represents the identity element. ( a ) showstransitions when the sixth card is chosen and ( b ) shows transitions when thelast card is chosen.The flip-transpose top with random shuffle on the hyperoctahedral group serves the casewhen | G n | = 2 for all n [10]. We now state the main theorems of this paper. Theorem 1.1.
The warp-transpose top with random shuffle on G n presents ℓ -pre-cutoff at n log n + n log( | G n | − . Moreover this shuffle has ℓ -cutoff at n log n + n log( | G n | − if | b G n | = o ( | G n | δ n δ ) for all δ > . Theorem 1.2.
The mixing time for the warp-transpose top with random shuffle on G n is O (cid:16) n log n + n log( | G n | − (cid:17) . Theorem 1.3.
The warp-transpose top with random shuffle on G n exhibits total variationcutoff phenomenon with cutoff time n log n + n log( | G n | − if | G n | = o ( n δ ) for all δ > . Let us recall some concepts and terminologies from representation theory of finite groupand discrete time Markov chains with finite state space to make this paper self contained.Readers from representation theoretic background may skip Subsection 1.1 and from Prob-abilistic background may skip Subsection 1.2.1.1.
Representation theory of finite group.
Let G be a finite group and V be a finitedimensional complex vector space. Also let GL ( V ) be the set of all invertible linear operatorson V . A linear representation ρ of G is a homomorphism from G to GL ( V ). Sometimesthis representation is also denoted by the pair ( ρ, V ). The dimension of the vector space V is called the dimension d ρ of the representation. V is called the G -module corresponding tothe representation ρ in this case. Let C [ G ] be the group algebra consists of complex linear SUBHAJIT GHOSH combinations of elements of G . In particular taking V = C [ G ], we define the right regularrepresentation R : G −→ GL ( C [ G ]) of G by R ( g ) X h ∈ G C h h = X h ∈ G C h hg, where C h ∈ C . A vector subspace W of V is said to be stable ( or ‘ invariant ’) under ρ if ρ ( g ) ( W ) ⊂ W forall g in G . The representation ρ is irreducible if V is non-trivial and V has no non-trivialproper stable subspace. Two representations ( ρ , V ) and ( ρ , V ) of G are are said to be isomorphic if there exists an invertible linear map T : V → V such that the followingdiagram commutes for all g ∈ G : V V V V ρ ( g ) T Tρ ( g ) For each g ∈ G, ρ ( g ) can also be thought of as an invertible complex matrix of size d ρ × d ρ .The trace of the matrix ρ ( g ) is said to be the character value of ρ at g and is denoted by χ ρ ( g ). It can be easily seen that the character values are constants on conjugacy classes,hence characters are class functions. If χ ρ ( g ) denote the complex conjugate of χ ρ ( g ), thenone can check that χ ρ ( g − ) = χ ρ ( g ) for all g ∈ G . Let C ( G ) be the complex vector space ofclass functions of G . Then a ‘standard’ inner product h· , ·i on C ( G ) is defined as follows: h φ, ψ i = 1 | G | X g ∈ G φ ( g ) ψ ( g − ) for φ, ψ ∈ C ( G ) . An important theorem in this context is the following [21, Theorem 6]:
The characters corre-sponding to the non-isomorphic irreducible representations of G forms an h· , ·i -orthonormalbasis of C ( G ).If V ⊗ V denotes the tensor product of the vector spaces V and V , then the tensorproduct of two representations ρ : G → GL ( V ) and ρ : G → GL ( V ) is a representationdenoted by ( ρ ⊗ ρ , V ⊗ V ) and defined by,( ρ ⊗ ρ )( g )( v ⊗ v ) = ρ ( g )( v ) ⊗ ρ ( g )( v ) for v ∈ V , v ∈ V and g ∈ G. We use some results from representation theory of finite groups without recalling the proof.For details about finite group representation see [16, 18, 21].1.2.
Discrete time Markov chain with finite state space.
Let Ω be a finite set. Asequence of random variables X , X , . . . is a discrete time Markov chain with state space Ω and transition matrix M if for all x, y ∈ Ω, all k >
1, and all events H k − := ∩ ≤ s
0, we have(2) P ( X k +1 = y | H k − ∩ { X k = x } ) = M ( x, y ) . Equation (2) says that given the present, the future is independent of the past. Let D k denotethe distribution after k transitions, i.e. D k is the row (probability) vector ( P ( X k = x )) x ∈ Ω .Then D k = D k − M for all k ≥
1, which implies D k = D M k . In particular if the chain UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 5 starts at x ∈ Ω, then its distribution after k transitions is D k = δ x M k , i.e. P ( X k = y | X = x ) = M k ( x, y ). Here δ x is defined on Ω as follows: δ x ( u ) = u = x, u = x. A Markov chain is said to be irreducible if it is possible for the chain to reach any statestarting from any state using only transitions of positive probability. The period of a state x ∈ Ω is defined to be the greatest common divisor of the set of all times when it is possiblefor the chain to return to the starting state x . The period of all the states of an irreducibleMarkov chain are the same [13, Lemma 1.6]. An irreducible Markov chain is said to be aperiodic if the common period for all its states is 1. A probability distribution Π is said tobe a stationary distribution of the Markov chain if Π M = Π. Any irreducible Markov chainpossesses a unique stationary distribution Π with Π( x ) > x ∈ Ω [13, Proposition1.14]. Moreover if the chain is aperiodic then D k −→ Π as k −→ ∞ [13, Theorem 4.9]. For anirreducible chain, we first define the ℓ -distance between the distribution after k transitionsand the stationary distribution. Definition 1.2.
Let D k denote the distribution after k transitions of an irreducible discretetime Markov chain with finite state space Ω, and Π denote its stationary distribution. Thenthe ℓ -distance between D k and Π is defined by || D k − Π || := X x ∈ Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D k ( x )Π( x ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Π( x ) . We now define the total variation distance between two probability measures.
Definition 1.3.
Let µ and ν be two probability measures on Ω. The total variation distance between µ and ν is defined by || µ − ν || TV := sup A ⊂ Ω | µ ( A ) − ν ( A ) | . It can be easily seen that || µ − ν || TV = P x ∈ Ω | µ ( x ) − ν ( x ) | (see [13, Proposition 4.2]).For an irreducible and aperiodic chain the interesting topic is the minimum number oftransitions k required to reach near the stationarity Π up to a certain level of tolerance ε > ℓ -distance (respectively total variation distance) between thedistribution after k transitions and the stationary distribution as follows: d ( k ) := max x ∈ Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k ( x, · ) − Π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) respectively d TV ( k ) := max x ∈ Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k ( x, · ) − Π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV (cid:19) . For ε >
0, the ℓ -mixing time (respectively total variation mixing time) with tolerance level ε is defined by τ mix ( ε ) := min { k : d ( k ) ≤ ε } (respectively t mix ( ε ) := min { k : d TV ( k ) ≤ ε } ) . Most of the notations of this subsection are borrowed from [13].
SUBHAJIT GHOSH
Non-commutative Fourier analysis and random walks on finite groups.
Let p and q be two probability measures on a finite group G . We define the convolution p ∗ q of p and q by ( p ∗ q )( x ) := P y ∈ G p ( xy − ) q ( y ). The Fourier transform b p of p at the right regularrepresentation R is defined by the matrix P x ∈ G p ( x ) R ( x ). The matrix b p ( R ) can be thoughtof as the action of the group algebra element P g ∈ G p ( g ) g on C [ G ] by multiplication on theright. It can be easily seen that \ ( p ∗ q )( R ) = b p ( R ) b q ( R ).A random walk on a finite group G driven by a probability measure p is a Markov chainwith state space G and transition probabilities M p ( x, y ) = p ( x − y ), x, y ∈ G . It can beeasily seen that the transition matrix M p is the transpose of b p ( R ) and the distribution after k th transition will be p ∗ k (convolution of p with itself k times) i.e., the probability of gettinginto state y starting at state x after k transitions is p ∗ k ( x − y ). One can easily check that therandom walk on G driven by p is irreducible if and only if the support of p generates G [19,Proposition 2.3]. The stationary distribution for an irreducible random walk on G driven by p , is the uniform distribution U G on G (since P x ∈ G M p ( x, y ) = P x ∈ G p ( x − y ) = P z ∈ G p ( z ) =1 , z = x − y for all y ∈ G ). From now on, the uniform distribution on group G will bedenoted by U G . For the random walk on G driven by p , it is enough to focus on (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV because, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M kp ( x, · ) − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M kp ( y, · ) − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M kp ( x, · ) − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M kp ( y, · ) − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for any two elements x, y ∈ G . We now define the cutoff and pre-cutoff phenomenon for asequence of random walks on finite groups. Definition 1.4.
Let { G n } ∞ be a sequence of finite groups and p n be a probability measureon G n for each n . Consider the sequence of irreducible and aperiodic random walk on G n driven by p n . We say that the ℓ -cutoff phenomenon (respectively total variation cutoffphenomenon ) holds for the family { ( G n , p n ) } ∞ if there exists a sequence { T n } ∞ of positivereal numbers tending to infinity as n → ∞ , such that the following hold:(1) For any ǫ ∈ (0 ,
1) and k n = ⌊ (1 + ǫ ) T n ⌋ ,lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (cid:18) respectively lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = 0 (cid:19) , (2) For any ǫ ∈ (0 ,
1) and k n = ⌊ (1 − ǫ ) T n ⌋ ,lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∞ (cid:18) respectively lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = 1 (cid:19) . Here ⌊ x ⌋ denotes the floor of x (the largest integer less than or equal to x ).Informally, we will say that { ( G n , p n ) } ∞ has an ℓ -cutoff (respectively total variation cutoff)at time T n . This says that for sufficiently large n the mixing time does not depend on thetolerance level ε ( > k transitions is very close tothe stationary distribution if k = T n but too far from the stationary distribution if k < T n .Roughly the cutoff phenomenon depends on the multiplicity of the second largest eigenvalueof the transition matrix [6]. UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 7
Definition 1.5.
Let { G n } ∞ be a sequence of finite groups and p n be a probability measureon G n for each n . Consider the sequence of irreducible and aperiodic random walk on G n driven by p n . We say that the family { ( G n , p n ) } ∞ presents an ℓ -pre-cutoff (respectivelytotal variation pre-cutoff) if there exists a sequence { T n } ∞ of positive real numbers tendingto infinity as n → ∞ , and two constants 0 < a < b < ∞ such that the following hold:(1) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ b T n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (cid:18) respectively lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ b T n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = 0 (cid:19) ,(2) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ a T n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∞ or positive (cid:18) respectively lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ a T n n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV > (cid:19) .We now see that the random walk of our concern is irreducible and aperiodic. Proposition 1.4.
The warp-transpose top with random shuffle on G n is irreducible andaperiodic.Proof. The support of P is Γ = { ( g − ) ( i ) g ( n ) ( i, n ) , g ( n ) | g ∈ G n , ≤ i < n } and it can beeasily seen that { g ( k ) , ( i, n ) | g ∈ G n , ≤ k ≤ n, ≤ i < n } is a generating set of G n .( g − ) ( n ) (cid:16) ( g − ) ( i ) g ( n ) ( i, n ) (cid:17) g ( n ) = ( i, n ) for each 1 ≤ i < n and g ∈ G n , ( k, n ) g ( n ) ( k, n ) = g ( k ) for each 1 ≤ k ≤ n and for all g ∈ G n . (3)Thus (3) implies Γ generates G n and hence the warp-transpose top with random shuffle on G n is irreducible. Moreover given any π ∈ G n , the set of all times when it is possible for thechain to return to the starting state π contains the integer 1 (as support of P contains theidentity element of G n ). Therefore the period of the state π is 1 and hence from irreducibilityall the states of this chain have period 1. Thus this chain is aperiodic. (cid:3) Proposition 1.4 says that the warp-transpose top with random shuffle on G n converges tothe uniform distribution U G n as the number of transitions goes to infinity. In Section 2 wewill find the spectrum of b P ( R ). We will prove Theorems 1.1 and 1.2 in Section 3. In Section4, lower bound of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV will be discussed and Theorem 1.3 will be proved. Acknowledgement.
I extend sincere thanks to my advisor Arvind Ayyer for all the insight-ful discussions during the preparation of this paper. I would like to thank the anonymousreviewer of FPSAC 2019 for the suggestion to extend the work from the S ≀ S n to G ≀ S n . Iwould like to acknowledge support in part by a UGC Centre for Advanced Study grant.2. Spectrum of the transition matrix
In this section we find the eigenvalues of the transition matrix b P ( R ), the Fourier transformof P at the right regular representation R of G n . To find the eigenvalues of b P ( R ) we willuse the representation theory of the wreath product G n of G n with the symmetric group S n .First we briefly discuss the representation theory of G ≀ S n , following the notation from [14].A partition λ of a positive integer n (denoted λ ⊢ n ) is a weakly decreasing finite sequence( λ , · · · , λ r ) of positive integers such that P ri =1 λ i = n . The partition λ can be pictoriallyvisualized as a left-justified arrangement of r rows of boxes with λ i boxes in the i th row,1 ≤ i ≤ r . This pictorial arrangement of boxes is known as the Young diagram of λ . For SUBHAJIT GHOSH example there are five partitions of the positive integer 4 viz. (4), (3 , , , ,
1) and(1 , , , Definition 2.1.
Let Y denote the set of all Young diagrams (there is a unique Youngdiagram with zero boxes) and Y n denote the set of all Young diagrams with n boxes. Forexample, elements of Y are shown in Figure 2. For a finite set X , we define Y (X) = { µ : µ is a map from X to Y } . For µ ∈ Y (X), define || µ || = P x ∈ X | µ ( x ) | , where | µ ( x ) | is thenumber of boxes of the Young diagram µ ( x ) and define Y n (X) = { µ ∈ Y (X) : || µ || = n } .(4) (3 ,
1) (2 ,
2) (2 , ,
1) (1 , , , Figure 2.
Young diagrams with 4 boxes.
Figure 3.
Standard Young tableaux of shape (3 , n be a fixed positive integer. Let b G denote the (finite) set of all non-isomorphic irre-ducible representations of G . Given σ ∈ b G , we denote by W σ the corresponding irreducible G -module (the space for the corresponding irreducible representation of G ). Elements of Y ( b G ) are called Young G -diagrams and elements of Y n ( b G ) are called Young G -diagrams with n boxes . Given µ ∈ Y ( b G ) and σ ∈ b G , we denote by µ ↓ σ the set of all Young G -diagramsobtained from µ by removing one of the inner corners in the Young diagram µ ( σ ). Let µ ∈ Y .A Young tableau of shape µ is obtained by taking the Young diagram µ and filling its | µ | boxes (bijectively) with the numbers 1 , , . . . , | µ | . A Young tableau is said to be standard if the numbers in the boxes strictly increase along each row and each column of the Youngdiagram of µ . The set of all standard Young tableaux of shape µ is denoted by tab( µ ).Elements of tab((3 , µ ∈ Y ( b G ). A Young G -tableau of shape µ is obtained by taking the Young G -diagram µ and filling its || µ || boxes (bijectively) withthe numbers 1 , , . . . , || µ || . A Young G -tableau is said to be standard if the numbers in theboxes strictly increase along each row and each column of all Young diagrams occurring in µ . Let tab G ( n, µ ), where µ ∈ Y n ( b G ), denote the set of all standard Young G -tableaux ofshape µ and let tab G ( n ) = ∪ µ ∈Y n ( b G ) tab G ( n, µ ). Let T ∈ tab G ( n ) and i ∈ [ n ]. If i appear inthe Young diagram µ ( σ ), where µ is the shape of T and σ ∈ b G , we write r T ( i ) = σ . The content of a box in row p and column q of a Young diagram is the integer q − p . Let b T ( i ) bethe box in µ ( σ ), with the number i resides and c ( b T ( i )) denote the content of the box b T ( i ). UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 9
For example let us take n = 10 and G to be Z , the additive group of integers modulo10. Also let b Z := { σ , σ , σ , σ , σ , σ , σ , σ , σ , σ } and µ ∈ Y ( b Z ) be such that µ ( σ ) = , µ ( σ ) = , µ ( σ ) = , µ ( σ ) =and µ ( σ i ) = φ for all i ∈ { , , , , , } , where φ denotes the empty Young diagram (i.e.Young diagram with no boxes). Then for the element T of tab Z (10 , µ ) given by µ ( σ ) , µ ( σ ) , µ ( σ ) , µ ( σ ) and µ ( σ i ) φ for i ∈ { , , , , , } , we have the following: r T (1) = σ , r T (2) = σ , r T (3) = σ , r T (4) = σ , r T (5) = σ , r T (6) = σ , r T (7) = σ , r T (8) = σ , r T (9) = σ , r T (10) = σ and c ( b T (1)) = 0, c ( b T (2)) = − c ( b T (3)) = 0, c T (4) = 0, c T (5) = 0, c ( b T (6)) =1, c ( b T (7)) = − c ( b T (8)) = − c ( b T (9)) = 2, c ( b T (10)) = 0. Definition 2.2.
Let H i,n ( G ) be the subgroup { ( g , . . . , g n , π ) ∈ G ≀ S n : π ( j ) = j for i + 1 ≤ j ≤ n } of G ≀ S n for 0 ≤ i ≤ n . In particular H ,n ( G ) = H ,n ( G ) = G n and H n,n ( G ) = G ≀ S n . Definition 2.3.
The (generalized)
Young-Jucys-Murphy elements X ( G ) , . . . , X n ( G ) of H n,n ( G ) or C [ G ≀ S n ] are given by X ( G ) = 0 and X i ( G ) = i − X k =1 X g ∈ G ( g − ) ( k ) g ( i ) ( k, i ) = i − X k =1 X g ∈ G ( g − ) ( k ) ( k, i ) g ( k ) , for all 2 ≤ i ≤ n. Young-Jucys-Murphy elements generates a maximal commuting subalgebra of C [ G ≀ S n ]and act like scalars on the Gelfand-Tsetlin subspaces of irreducible G ≀ S n -modules. We nowdefine Gelfand-Tsetlin subspaces and the Gelfand-Tsetlin decomposition.Let λ ∈ d H n,n ( G ) and consider the irreducible H n,n ( G )-module (the space for the represen-tation of H n,n ( G )) V λ . Since the branching is simple [14, Section 4], the decomposition intoirreducible H n − ,n ( G )-modules is given by V λ = ⊕ µ V µ , where the sum is over all µ ∈ \ H n − ,n ( G ), with µ ր λ (i.e there is an edge from µ to λ inthe branching multi-graph), is canonical. Iterating this decomposition of V λ into irreducible H ,n ( G )-submodules, i.e.,(4) V λ = ⊕ T V T , where the sum is over all possible chains T = λ ր λ ր · · · ր λ n with λ i ∈ d H i,n ( G )and λ n = λ . We call (4) the Gelfand-Tsetlin decomposition of V λ and each V T in (4) a Gelfand-Tsetlin subspace of V λ . We note that if 0 = v T ∈ V T , then C [ H i,n ( G )] v T = V λ i from the definition of V T . From Lemma 6.2 and Theorem 6.4 of [14], we may parametrisethe irreducible representations of G ≀ S n by elements of Y n ( b G ). Theorem 2.1 ([14, Theorem 6.5]) . Let µ ∈ Y n ( b G ) . Then we may index the Gelfand-Tsetlinsubspaces of V µ by standard Young G -tableaux of shape µ and write the Gelfand-Tsetlindecomposition as V µ = ⊕ T ∈ tab G ( n,µ ) V T , where each V T is closed under the action of G n and as a G n -module, is isomorphic to theirreducible G n -module W r T (1) ⊗ W r T (2) ⊗ · · · ⊗ W r T ( n ) . For i = 1 , . . . , n ; the eigenvalues of X i ( G ) on V T are given by | G | dim( W rT ( i ) ) c ( b T ( i )) . Theorem 2.2 ([14, Theorem 6.7]) . Let µ ∈ Y n ( b G ) . Write the elements of b G as { σ , . . . , σ t } and set µ ( i ) = µ ( σ i ) , m i = | µ ( i ) | , d i = dim( W σ i ) for each ≤ i ≤ t . Then dim( V µ ) = nm , . . . , m t ! f µ (1) · · · f µ ( t ) d m · · · d m t t . Here f µ ( i ) denotes the number of standard Young tableau of shape µ ( i ) , for each ≤ i ≤ t . Lemma 2.3.
Let G be a finite group and σ ∈ b G . If W σ ( respectively χ σ ) denotes theirreducible G -module ( respectively character ) and d σ is the dimension of W σ , then the actionof the group algebra element P g ∈ G g on W σ is given by the following scalar matrix X g ∈ G g = | G | d σ h χ σ , χ i I d σ . Here I d σ is the identity matrix of order d σ × d σ and be the trivial representation of G .Proof. It is clear that P g ∈ G g is in the centre of C [ G ]. Therefore by Schur’s lemma ([21,Proposition 4]), we have P g ∈ G g = cI d σ for some c ∈ C . The value of c can be obtained byequating the traces of P g ∈ G g and cI d σ . (cid:3) Remark 2.4.
Our focus will be on H n,n ( G n ) i.e. G n ≀ S n for the sequence of subgroups H ,n ( G n ) ⊆ · · · ⊆ H i,n ( G n ) ⊆ · · · ⊆ H n,n ( G n ) . For simplicity we write the Young-Jucys-Murphy elements X i ( G n ) of G n ≀ S n (i.e. G n ) as X i for ≤ i ≤ n . Thus Theorems 2.1 and 2.2 are applicable to G n . Let t := | b G n | and b G n := { σ , . . . , σ t } , where σ = (the trivial representation of G n ).We write µ (cid:16) ∈ Y n ( b G n ) (cid:17) as the tuple ( µ (1) , . . . , µ ( t ) ), where µ ( i ) := µ ( σ i ) for each 1 ≤ i ≤ t .We also denote m i := | µ ( i ) | , W σ i := the irreducible G n -module corresponding to σ i and d i = dim( W σ i ) for each 1 ≤ i ≤ t . Thus t, σ i , µ ( i ) , m i , W σ i , and d i depend on G n i.e., on n . To avoid notational complication the dependence of t, σ i , µ ( i ) , m i , W σ i , and d i on n issuppressed. We note that for T ∈ tab G n ( n, µ ) the dimension of V T is d m · · · d m t t . Theorem 2.5.
For each µ = ( µ (1) , . . . , µ ( t ) ) ∈ Y n ( b G n ) , let b P ( R ) (cid:12)(cid:12)(cid:12) V µ denote the restriction of b P ( R ) to the irreducible G n -module V µ . Then the eigenvalues of b P ( R ) (cid:12)(cid:12)(cid:12) V µ are given by, n dim( W r T ( n ) ) (cid:16) c ( b T ( n )) + h χ r T ( n ) , χ i (cid:17) , with multiplicity dim( V T ) = d m · · · d m t t UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 11 for each T ∈ tab G n ( n, µ ) .Proof. We first find the eigenvalues of X n + P g ∈ G n (e , . . . , e , g ; id). Let I dim( V T ) denote theidentity matrix of order dim( V T ) × dim( V T ). Then from Theorem 2.1 we have(5) V µ = ⊕ T ∈ tab Gn ( n,µ ) V T and X n (cid:12)(cid:12)(cid:12) V T = | G n | dim( W r T ( n ) ) c ( b T ( n )) I dim( V T ) . Again from Theorem 2.1 and Lemma 2.3 we have(6) X g ∈ G n (e , . . . , e , g ; id) (cid:12)(cid:12)(cid:12) V T = | G n | dim( W r T ( n ) ) h χ r T ( n ) , χ i I dim( V T ) . We recall b P ( R ) = 1 n | G n | X g ∈ G n R ((e , . . . , e , g ; id)) + n − X i =1 R (cid:16) (e , . . . , e , g − , e , . . . , e , g ; ( i, n )) (cid:17)! . Therefore n | G n | b P ( R ) is the action of X n + P g ∈ G n (e , . . . , e , g ; id) on C [ G n ] by multiplicationon the right. Since dim( V T ) = d m · · · d m t t , the theorem follows from (5) and (6). (cid:3) Remark 2.6.
In the regular representation of a finite group, each irreducible representationoccurs with multiplicity equal to its dimension [21, section 2.4] . Therefore, Theorems 2.2 and2.5 provide the eigenvalues of b P ( R ) . Upper bound for total variation distance
In this section, we find upper bounds of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV when k ≥ n log n + n log( | G n | −
1) + n log( | b G n | −
1) + Cn for C >
0. We also prove Theorems 1.1and 1.2 in this section. Before proving the main results of this section, first we set somenotations and prove two useful lemmas. For any positive integer N, we write ξ ⊢ N to denote ξ is a partition of N. Given a partition ξ of the integer N (here we are allowing N to takevalue 0), throughout this section ξ denotes the largest part of ξ . In particular if ξ ⊢ f ξ = 1 (as there is a unique Young diagram with zero boxes) and we set ξ = 0. Theorem 3.1 (Plancherel formula, [4, Theorem 4.1]) . Let f and f be two functions on thefinite group G. Then X g ∈ G f ( g − ) f ( g ) = 1 | G | X ρ ∈ b G d ρ trace (cid:16) ˆ f ( ρ ) ˆ f ( ρ ) (cid:17) , where the sum is over all irreducible representations ρ of G and d ρ is the dimension of ρ . Recall that U G be the uniform distribution on the group G . Then using Lemma 2.3 wehave the following b U G ( ρ ) = ρ = , ρ = , for ρ ∈ b G. Moreover, given any probability measure p on the finite group G , we have b p ( ) = 1. Thereforesetting f = f = p ∗ k − U G , we have the following(7) p ( x ) = p ( x − ) for all x ∈ G = ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X ρ ∈ b G \{ } d ρ trace (cid:16) ( b p ( ρ )) k (cid:17) . We now state the Diaconis-Shahshahani upper bound lemma. The proof follows from theCauchy-Schwarz inequality and (7).
Lemma 3.2 ([4, Lemma 4.2]) . Let p be a probability measure on a finite group G such that p ( x ) = p ( x − ) for all x ∈ G . Suppose the random walk on G driven by p is irreducible. Thenwe have the following (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∗ k − U G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 14 X ρ ∈ b G \{ } d ρ trace (cid:16) ( b p ( ρ )) k (cid:17) , where the sum is over all non-trivial irreducible representations ρ of G and d ρ is the dimen-sion of ρ . Definition 3.1.
Let A be a non empty set. Then the indicator function of A is denoted by Ind A and is defined by Ind A ( x ) = x ∈ A x / ∈ A. Lemma 3.3.
Let N be a positive integer and s be any non-negative real number. Then wehave X λ ⊢ N ( f λ ) λ − s N ! k < e − ks N e N e − k N . Proof.
For ζ ⊢ (N − λ ), recall that ζ denotes the largest part of ζ . If ζ ≤ λ , then we have f λ ≤ (cid:16) N λ (cid:17) f ζ . Therefore X λ ⊢ N ( f λ ) λ − s N ! k is less than or equal to N X λ =1 X ζ ⊢ (N − λ ) ζ ≤ λ N λ ! ( f ζ ) λ − s N ! k ≤ N X λ =1 N λ ! λ − s N ! k X ζ ⊢ (N − λ ) ( f ζ ) = N − X u =0 N u ! (cid:18) − u + s N (cid:19) k u ! . (8)Equality in (8) is obtained by writing u = N − λ . Using 1 − x ≤ e − x for all x ≥ (cid:16) N u (cid:17) ≤ N u u ! , the expression in the right hand side of (8) is less than or equal to N − X u =0 N u u ! e − k N ( u + s ) < e − ks N ∞ X u =0 u ! (cid:16) N e − k N (cid:17) u = e − ks N e N e − k N . (cid:3) An immediate corollary of Lemma 3.3 follows from the fact (cid:16) f λ (cid:17) λ − s N ! k = N − s N ! k , if λ = (N) ⊢ N . Corollary 3.4.
Following the notations of Lemma 3.3, we have X λ ⊢ N λ =( N ) ( f λ ) λ − s N ! k < e − ks N e N e − k N − N − s N ! k . UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 13
Lemma 3.5.
Let µ = ( µ (1) , . . . , µ ( t ) ) ∈ Y n ( b G n ) . Recall that µ ( j )1 ( respectively µ ( j ) ′ ) denotesthe largest part of µ ( j ) ( respectively its conjugate ) for ≤ j ≤ t . Then we have X T ∈ tab Gn ( n,µ ) c ( b T ( n )) + h χ r T ( n ) , χ i n dim( W r T ( n ) ) ! k < nm , . . . , m t ! f µ (1) · · · f µ ( t ) t X j =1 (cid:16) M kj + M ′ kj (cid:17) Ind (0 , ∞ ) ( m j ) , where M j := µ ( j )1 − h χ σj ,χ i nd j and M ′ j := µ ( j ) ′ − h χ σj ,χ i nd j for each ≤ j ≤ t .Proof. Let T i = { ( T , . . . , T t ) ∈ tab G n ( n, µ ) | b T ( n ) is in T i } for each 1 ≤ i ≤ t . Thentab G n ( n, µ ) is the disjoint union of the sets T , . . . , T t . Therefore we have X T ∈ tab Gn ( n,µ ) c ( b T ( n )) + h χ r T ( n ) , χ i n dim( W r T ( n ) ) ! k = t X i =1 X T ∈T i c ( b T ( n )) + h χ σ i , χ i nd i ! k Ind (0 , ∞ ) ( m i )and this is equal to, t X i =1 n − m , .., m i − , .., m t ! f µ (1) · · · f µ ( t ) f µ ( i ) X T i ∈ tab( µ ( i ) ) c ( b T i ( m i )) + h χ σ i , χ i nd i ! k Ind (0 , ∞ ) ( m i ) < t X i =1 nm , . . . , m t ! f µ (1) · · · f µ ( t ) f µ ( i ) X T i ∈ tab( µ ( i ) ) (cid:16) M ki + M ′ ki (cid:17) Ind (0 , ∞ ) ( m i ) . (9)The inequality in (9) holds because T i ∈ tab( µ ( i ) ) implies the following: c ( b T i ( m i )) + h χ σ i , χ i nd i ! k ≤ max µ ( i )1 − h χ σ i , χ i nd i k , µ ( i ) ′ − − h χ σ i , χ i nd i k ≤ max µ ( i )1 − h χ σ i , χ i nd i k , µ ( i ) ′ − h χ σ i , χ i nd i k , as h χ σ i , χ i = 0 or 1 < µ ( i )1 − h χ σ i , χ i nd i k + µ ( i ) ′ − h χ σ i , χ i nd i k = M ki + M ′ ki . Therefore the result follows from (9) and X T i ∈ tab( µ ( i ) ) (cid:16) M ki + M ′ ki (cid:17) = f µ ( i ) (cid:16) M ki + M ′ ki (cid:17) . (cid:3) Proposition 3.6.
For the warp-transpose top with random shuffle on G n , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) e n e − kn − (cid:19) + e − kn + 2 e (cid:18) e n ( | G n |− e − kn − (cid:19) +2( | b G n | − (cid:18) e − kn e n e − kn + en (cid:18) e n ( | G n |− e − kn − (cid:19)(cid:19) , for all k ≥ max { n, n log n } .Proof. Let us recall that b G n = { σ , . . . , σ t } and σ = , the trivial representation of G n .Given µ ∈ Y n ( b G n ), throughout this proof we write µ = ( µ (1) , . . . , µ ( t ) ), where µ ( i ) = µ ( σ i ), µ ( i ) ⊢ m i , and P ti =1 m i = n . Now using Lemma 3.2, we have(10) 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X µ ∈Y n ( b G n ): µ ( ) =( n ) dim( V µ ) trace (cid:18)(cid:16) b P ( R ) (cid:12)(cid:12)(cid:12) V µ (cid:17) k (cid:19) . First we partition the set Y n ( b G n ) into two disjoint subsets A and A as follows: A = ∪ ≤ i ≤ t B i , where B i = { µ ∈ Y n ( b G n ) | m i = n, m k = 0 for all k ∈ [ t ] \ { i }}A = { µ ∈ Y n ( b G n ) | t X k =1 m k = n, ≤ m k ≤ n − } . It can be easily seen that B i ’s are disjoint. Therefore by using Theorem 2.5 and Remark 2.6,the inequality (10) become4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X µ ∈B µ ( ) =( n ) dim( V µ ) X T ∈ tab Gn ( n,µ ) c ( b T ( n )) + 1 nd ! k d n + t X i =2 X µ ∈B i dim( V µ ) X T ∈ tab Gn ( n,µ ) c ( b T ( n )) nd i ! k d ni (11) + X µ ∈A dim( V µ ) X T ∈ tab Gn ( n,µ ) c ( b T ( n )) + h χ r T ( n ) , χ i n dim( W r T ( n ) ) ! k d m · · · d m t t . The first two terms in the right hand side of (11) are equal to X λ ⊢ nλ = n f λ d n X T ∈ tab( λ ) c ( b T ( n )) + 1 nd ! k d n + t X i =2 X λ ⊢ n f λ d ni X T ∈ tab( λ ) c ( b T ( n )) nd i ! k d ni = X λ ⊢ nλ =( n ) , (1 n ) f λ X T ∈ tab( λ ) c ( b T ( n )) + 1 n ! k + (cid:18) n − n (cid:19) k + t X i =2 d ni d ki X λ ⊢ n f λ X T ∈ tab( λ ) c ( b T ( n )) n ! k . (12) UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 15
Now recalling λ (respectively λ ′ ) is the largest part of λ (respectively its conjugate), wehave the following: c ( b T ( n )) + xn ! k ≤ max λ − xn ! k , λ ′ − − xn ! k ,< λ − xn ! k + λ ′ − xn ! k , for T ∈ tab( λ ) and x ≥ . This implies X λ ⊢ nλ =( n ) , (1 n ) f λ X T ∈ tab( λ ) c ( b T ( n )) + 1 n ! k < X λ ⊢ nλ =( n ) , (1 n ) (cid:16) f λ (cid:17) λ n ! k + λ ′ n ! k < X λ ⊢ nλ =( n ) , (1 n ) (cid:16) f λ (cid:17) λ n ! k and X λ ⊢ n f λ X T ∈ tab( λ ) c ( b T ( n )) n ! k < X λ ⊢ n (cid:16) f λ (cid:17) λ − n ! k + λ ′ − n ! k < X λ ⊢ n (cid:16) f λ (cid:17) λ − n ! k . Thus using 1 − x ≤ e x for x ≥ k ≥ n , and d i ≥ ≤ i ≤ t , the expression in (12)is bounded above by2 X λ ⊢ nλ =( n ) ( f λ ) λ n ! k + (cid:18) − n (cid:19) k + 2 t X i =2 X λ ⊢ n ( f λ ) λ − n ! k < (cid:18) e n e − kn − (cid:19) + e − kn + 2( t − e − kn e n e − kn . (13)The inequality in (13) follows from Corollary 3.4 and Lemma 3.3. Now recalling M j := µ ( j )1 − h χ σj ,χ i nd j , M ′ j := µ ( j ) ′ − h χ σj ,χ i nd j , and using Lemma 3.5, the third term in the right handside of (11) is less than X µ ∈A nm , . . . , m t ! ( f µ (1) ) · · · ( f µ ( t ) ) d m . . . d m t t t X j =1 (cid:16) M kj + M ′ kj (cid:17) Ind (0 , ∞ ) ( m j ) . (14)We now deal with (14) by considering two separate cases namely j = 1 and 1 < j ≤ t . Nowusing X µ (1) ⊢ m (cid:16) f µ (1) (cid:17) µ (1) ′ nd k = X µ (1) ⊢ m (cid:16) f µ (1) (cid:17) µ (1)1 nd k , the partial sum corresponding to j = 1 in (14) is equal to, n − X m =1 X ( m ,...,m t ) P m k = n − m ≤ m k ≤ n − X µ ( i ) ⊢ m i ≤ i ≤ t nm ! n − m m , . . . , m t ! ( f µ (1) ) · · · ( f µ ( t ) ) d m . . . d m t t µ (1)1 nd k =2 n − X m =1 ( d + · · · + d t ) n − m nm ! ( n − m )! (cid:18) d (cid:19) k − m (cid:18) m n (cid:19) k X µ (1) ⊢ m ( f µ (1) ) µ (1)1 m k (15) < n − X m =1 ( d + · · · + d t ) n − m nm ! ( n − m )! (cid:18) d (cid:19) k − m (cid:18) m n (cid:19) k e m e − km . The inequality in (15) follows from Lemma 3.3. As k ≥ n log n , we have k ≥ m log m .Thus writing n − m by u , the expression in (15) is less than or equal to(16) 2 e n − X u =1 d + · · · + d t d ! u (cid:18) d (cid:19) k − n nu ! u ! (cid:18) − un (cid:19) k Now using 1 − x ≤ e − x for all x ≥ d = 1 the expression in (16) is less than or equalto(17) 2 e n − X u =1 u ! n | G n | d − ! e − kn ! u < e e (cid:16) n (cid:16) | Gn | d − (cid:17) e − kn (cid:17) − . Now using the notation m , .., d m j , .., m t to denote m , . . . , m j − , m j +1 , . . . , m t , and X µ ( j ) ⊢ m j (cid:16) f µ ( j ) (cid:17) µ ( j ) ′ nd j k = X µ ( j ) ⊢ m j (cid:16) f µ ( j ) (cid:17) µ ( j )1 nd j k , the partial sum corresponding to 1 < j ≤ t in (14) turns out to be(18) n − X m j =1 X ( m ,..., c m j ,...,m t ) P m k = n − m j ≤ m k ≤ n − X µ ( i ) ⊢ m i ≤ i ≤ t nm j ! n − m j m , . . . , d m j , . . . , m t ! ( f µ (1) ) · · · ( f µ ( t ) ) d m . . . d m t t ζ k , where ζ = µ ( j )1 − nd j . The expression given in (18) is equal to the following2 n − X m j =1 ( d + · · · + d t − d j ) n − m j nm j ! ( n − m j )! d j ! k − m j (cid:18) m j n (cid:19) k X µ ( j ) ⊢ m j ( f µ ( j ) ) µ ( j )1 − m j k (19) < n − X m j =1 ( d + · · · + d t − d j ) n − m j nm j ! ( n − m j )! d j ! k − m j (cid:18) m j n (cid:19) k e − kmj e m j e − kmj . UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 17
The inequality in (19) follows from Lemma 3.3. As k ≥ n log n , we have k ≥ m j log m j and k ≥ m j log n . Thus writing n − m j by v , the expression in (19) is less than or equal to(20) 2 en n − X v =1 d + · · · + d t − d j d j ! v d j ! k − n nv ! v ! (cid:18) − vn (cid:19) k Now using 1 − x ≤ e − x for all x ≥ d k − nj ≥ j ∈ { , . . . , t } , the expression in(20) is less than or equal to(21) 2 en n − X v =1 v ! n | G n | d j − ! e − kn ! v < en e (cid:18) n (cid:18) | Gn | d j − (cid:19) e − kn (cid:19) − . Therefore the proposition follows from (11), (13), (17), (21) and d j ≤ ≤ j ≤ t . (cid:3) Proposition 3.7.
For large n , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > q ( n − n ( | G n | − n − e − kn . Proof.
Recall that the irreducible representations of G n are parameterised by the elements of Y n ( b G n ). We now use Theorem 2.5 to compute the eigenvalues of the restriction of b P ( R ) tosome irreducible G n -modules. The eigenvalues of the restriction of b P ( R ) to the irreducible G n -module indexed by n − z }| { · · · , φ, . . . , φ ∈ Y n ( b G n )(22)are given below. Eigenvalues: 1 − n n − b P ( R ) to the irreducible G n -modules indexed by Young G n -diagram with n boxes of the following form n − z }| { · · · , φ, . . . , φ, ↑ , φ, . . . , φ ∈ Y n ( b G n ) , for 1 < i ≤ | b G n | i th position.(23)are given below. Eigenvalues: 1 − n n − d i d i Now (7) implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ( n − d n ( n − (cid:18) − n (cid:19) k ! + | b G n | X i =2 nd n − d i ( n − d i (cid:18) − n (cid:19) k ! ≈ ( n − n − e − kn + n ( n − | G n | − e − kn . (24) Here ‘ a n ≈ b n ’ means ‘ a n is asymptotic to b n ’ i.e. a n b n = 1 as n → ∞ . Therefore (24) implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > q ( n − n ( | G n | − n − e − kn for large n . (cid:3) Proof of Theorem 1.1.
Let us set k = 2 (cid:16) n log n + n log ( | G n | − (cid:17) . Then Proposition 3.6implies the following:0 ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ( n log n + n log( | G n |− ) − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) e n | Gn |− − (cid:19) + 1 n ( | G n | − + 2 e (cid:18) e n | Gn |− − (cid:19) + 2 (cid:16) | b G n | − (cid:17) n ( | G n | − e n | Gn |− + en (cid:18) e n | Gn |− − (cid:19)! < n ( | G n | − + 1 n ( | G n | − + 4 en ( | G n | − (cid:16) | b G n | − (cid:17) n ( | G n | − e n | Gn |− + 2 en ( | G n | − ! . The inequality in (25) follows from the fact that e x − < x for all 0 < x ≤
1. Thereforethe right hand side of (25) converges to zero as n → ∞ implies that(26) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ( n log n + n log( | G n |− ) − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Now for large n , setting k = n log n + n log ( | G n | −
1) in Proposition (3.7), we have thefollowing (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ( n log n + n log( | G n |− ) − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > vuut n − n ( | G n | − n ( | G n | − ! (cid:18) − n (cid:19) = ⇒ lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ( n log n + n log( | G n |− ) − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ lim n →∞ vuut n − n ( | G n | − ! (cid:18) − n (cid:19) = 1 . (27)Therefore (25) and (27) implies that the warp-transpose top with random shuffle on G n presents ℓ -pre-cutoff at n log n + n log ( | G n | − ǫ ∈ (0 , k n = j (1 + ǫ ) (cid:16) n log n + n log ( | G n | − (cid:17)k , we have k n + 1 ≥ (1 + ǫ ) (cid:18) n log n + 12 n log ( | G n | − (cid:19) = ⇒ e − knn ≤ e n n ǫ ( | G n | − ǫ . UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 19
Therefore Proposition 3.6 implies0 ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < e e nn ǫ ( | Gn |− ǫ − + e n n ǫ ( | G n | − ǫ + 2 e e e nn ǫ ( | Gn |− ǫ − + 2 (cid:16) | b G n | − (cid:17) e n n ǫ ( | G n | − ǫ e e nn ǫ ( | Gn |− ǫ + en e e nn ǫ ( | Gn |− ǫ − < e n n ǫ ( | G n | − ǫ + e n n ǫ ( | G n | − ǫ + 4 e n n ǫ ( | G n | − ǫ (28) + 2 (cid:16) | b G n | − (cid:17) e n n ǫ ( | G n | − ǫ e e nn ǫ ( | Gn |− ǫ + 2 e n n ǫ ( | G n | − ǫ . The inequality in (28) follows from the fact that e x − < x for all 0 < x ≤
1. If | b G n | = o ( n δ | G n | δ ) for all δ >
0, then the right hand side of (28) converges to zero as n → ∞ .Therefore we have the following:(29) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ⌊ (1+ ǫ ) ( n log n + n log( | G n |− ) ⌋ − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , if | b G n | = o ( n δ | G n | δ ) for all δ > . Again for 0 < ǫ <
1; setting k n = j (1 − ǫ ) (cid:16) n log n + n log ( | G n | − (cid:17)k , Proposition (3.7)implies the following(30) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ⌊ (1 − ǫ ) ( n log n + n log( | G n |− ) ⌋ − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > vuut n − n ( | G n | − ! (cid:18) − n (cid:19) n ǫ ( | G n | − ǫ , for large n . The right hand side of the inequality (30) tends to infinity as n → ∞ . Thereforefrom (29) and (30); we can conclude that if | b G n | = o ( n δ | G n | δ ) for all δ >
0, then thewarp-transpose top with random shuffle on G n satisfies the ℓ -cutoff phenomenon with cutofftime n log n + n log( | G n | − (cid:3) Lemma 3.8.
Let
C > . If k ≥ n log n + n log( | G n | −
1) + n log( | b G n | −
1) + Cn , then forthe random walk on G n driven by P we have the following: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < √ e + 1 e − C + o (1) . Proof.
Using k ≥ n log n + n log( | G n | −
1) + n log( | b G n | −
1) + Cn Proposition 3.6 we havethe following:4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < e e − C ( | Gn |− ( | b Gn |− ) − + e − C n ( | G n | − (cid:16) | b G n | − (cid:17) + 2 e e e − C ( | b Gn |− ) − (31) + 2( | b G n | − e − C n ( | G n | − (cid:16) | b G n | − (cid:17) e e − C ( | Gn |− ( | b Gn |− ) + en e e − C ( | b Gn |− ) − . The sequence { G n } ∞ consists of non-trivial finite groups, thus | G n |− ≤ | b G n |− ≤ e x − < x for 0 < x ≤
1, the expression in the right hand side of (31) isless than(2 + 2 e ) (cid:16) e e − C − (cid:17) + e − C n + 2 e − C n ( | G n | − e e − C + 2 (cid:16) | b G n | − (cid:17) × en × e − C ( | b G n | − e ) (cid:16) e e − C − (cid:17) + o (1) . Again using e x − < x for 0 < x ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < e ) e − C + o (1)= ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < √ e + 1 e − C + o (1) . (cid:3) Proof of Theorem 1.2.
Let 0 < ǫ < τ ( n )mix ( ǫ ) (respectively t ( n )mix ( ǫ )) be the ℓ -mixing time(respectively total variation mixing time) with tolerance level ǫ for the warp-transpose topwith random shuffle on G n . First choose C ǫ > √ e e − C ǫ < ǫ . The Lemma 3.8ensures the existence of positive integer N such that the following hold for all n ≥ N , k ≥ n log n + 12 n log( | G n | −
1) + 12 n log( | b G n | −
1) + C ǫ n = ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV < ǫ and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ. Therefore for all n ≥ N , using n log n + 12 n log( | G n | −
1) + 12 n log( | b G n | −
1) + C ǫ n < (cid:18) n log n + 12 n log( | G n | − (cid:19) , we can conclude that τ ( n )mix ( ǫ ) ≤ (cid:18) n log n + 12 n log( | G n | − (cid:19) and t ( n )mix ( ǫ ) ≤ (cid:18) n log n + 12 n log( | G n | − (cid:19) . Thus the theorem follows. (cid:3) Lower bound for total variation distance
In this section, we will focus on the lower bound of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV and prove the Theorem1.3. To establish the theorem giving this lower bound we use tools from representation theoryof finite group. We also use the action of n th Young-Jucys-Murphy element of the symmetricgroup S n on irreducible S n -modules. Before going into the heart of this section let us brieflydiscuss the Vershik-Okounkov approach for the representation theory of S n [15]. As weknow the irreducible S n -modules are indexed by partitions of n let us take a partition λ of n and denote the corresponding S n -module (known as Specht module) by S λ . In case ofsymmetric group the Gelfand-Tsetlin subspaces are one dimensional and they forms a basisfor the corresponding Specht module, known as Gelfand-Tsetlin basis. The Gelfand-Tsetlinbasis vectors of S λ are indexed by the standard Young tableaux of shape λ . Moreover theYoung-Jucys-Murphy elements act on the the Gelfand-Tsetlin basis vectors like scalars. The UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 21 precise action of the i th Young-Jucys-Murphy element P i − u =1 ( u, i ) on the Gelfand-Tsetlinbasis vector s T of S λ is given by(32) i − X u =1 ( u, i ) ! s T = c ( b T ( i )) s T for all T ∈ tab( λ ) and 1 < i ≤ n. Now we define the representation ρ def of the symmetric group S n , known as the definingrepresentation of S n . Let C [ { , . . . , n } ] be the subspace of V spanned by { , . . . , n } . Then ρ def : S n → GL ( C [ { , . . . , n } ]) is defined by ρ def ( π ) X c i ∈ C c i i = X c i ∈ C c i π ( i ) . Lemma 4.1.
The eigenvalues of n − P u =1 ρ def (( u, n )) are given below: Eigenvalues: n − n − − n − Proof.
The defining representation of S n splits into two irreducible Specht modules S ( n ) (trivial) and S ( n − , with multiplicity one each [18, Example 2.1.8 and Theorem 2.11.2].Therefore the lemma follows from (32) and straightforward calculations. (cid:3) Lemma 4.2. [10, Lemma 4.6]
Let β i denote the matrix n − P u =1 ρ def (( u, n )) − ρ def (( i, n )) for all ≤ i < n . Then we have the following: (1) The matrices β i and β j are similar for i = j and i, j ∈ { , . . . , n − } . (2) For each i ∈ { , . . . , n − } , the eigenvalues of β i are the following: Eigenvalues: n − n − − n − Lemma 4.3.
The eigenvalues of n − P u =1 (cid:16) ρ def (( u, n )) ⊗ ρ def (( u, n )) (cid:17) are given as follows: Eigenvalues: n − n − − − n − n −
2) 3 n − n − n − n + 5 Proof.
The decomposition of ρ def ⊗ ρ def into irreducible Specht modules is given below (see[18, Example 2.1.8] and [12, Lemma 2.9.16]): ρ def ⊗ ρ def = 2 S ( n ) ⊕ S ( n − , ⊕ S ( n − , ⊕ S ( n − , , . Here the coefficient of irreducible Specht module denotes its multiplicity in ρ def ⊗ ρ def . Inthis case also the lemma follows from (32) and straightforward calculations. (cid:3) Now come back to the random walk of our concern. To start with we define an auxiliaryrepresentation R of G n and a random variable X on G n .Let V = C [ G n × [ n ]] be the complex vector space of all formal linear combinations ofelements of G n × [ n ] and GL ( V ) be the set of all invertible linear maps from V to itself. Wenow define the representation R : G n −→ GL ( V ) on the basis elements of V by R ( g , . . . , g n ; π ) (( h, i )) = (cid:16) g π ( i ) h, π ( i ) (cid:17) . The random variable X counts the number of fixed points of the action of R i.e. X is thecharacter χ R of R . Let E k ( X ) be the expectation and V k ( X ) be the variance of X withrespect to the probability measure P ∗ k on G n . Also E U ( X ) denotes the expectation of X withrespect to the uniform distribution on G n . Our goal is to compute E k ( X ) , V k ( X ) , E U ( X )and use probabilistic inequalities (viz. Chebychev’s and Markov’s inequality). Proposition 4.4.
Let E U ( X ) denote the expectation of X with respect to U G n . Then E U ( X ) = 1 .Proof. For notational simplicity let us denote A = G n × [ n ]. Also for a ∈ A , Fix G n ( a ) denotesthe set consisting of elements G n , which fixes a . Then | Fix G n ( a ) | = | G n | n − ( n − a ∈ A . Because if a = ( x, i ) then a is fixed by | G n | n − ( n − G n of the form:( ∗ , . . . , ∗ , e ↑ , ∗ , . . . , ∗ ; ˜ π ) ∈ G n such that ˜ π ( i ) = i.i th position.Here ∗ can be chosen independently from G n and for each of these choice ˜ π can be chosenfrom S n such that ˜ π ( i ) = i . Therefore using the definition of expectation we have E U ( X ) = X g ∈G n U G n ( g ) X ( g ) = 1 | G n | X g ∈G n χ R ( g ) = 1 | G n | n n ! X g ∈G n trace ( R ( g )) . Now the lemma follows from the following fact X g ∈G n trace ( R ( g )) = trace X g ∈G n R ( g ) = X a ∈ A | Fix G n ( a ) | = X a ∈ A | G n | n − ( n − | G n | n n ! . (cid:3) Let V + be the subspace of V spanned by { v , v , . . . , v n } , where v i s are defined as follows:v i = X g ∈ G n ( g, i ) , for 1 ≤ i ≤ n. Also let V − be the subspace of V spanned by { v i g | ≤ i ≤ n, g ∈ G n \ { e }} , where v i g s aredefined byv i g = v i −| G n | ( g, i ) = X h ∈ G n ( h, i ) − | G n | ( g, i ) , for 1 ≤ i ≤ n and g ∈ G n . It can be seen that V = V + ⊕ V − , and both V + and V are invariant under R . Now wedefine the following sub-representations of RR + : G n → GL ( V + ) defined by R + ( g ) = R ( g ) (cid:12)(cid:12)(cid:12) V + for all g ∈ G n , R − : G n → GL ( V − ) defined by R − ( g ) = R ( g ) (cid:12)(cid:12)(cid:12) V − for all g ∈ G n . Using R + and R − , X can be written as follows: X ( g ) = trace ( R ( g )) = trace (cid:16) R + ( g ) (cid:17) + trace (cid:16) R − ( g ) (cid:17) for all g ∈ G n . Unless otherwise stated from now on we have the following notational assumptions: • I n (respectively O n ) denotes the identity (respectively zero) matrix of order n × n . • M i denotes the matrix of order n × n with 1 at ( i, i )th position and 0 elsewhere. UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 23 • ρ def ( π ) denotes the matrix of its action on C [ { , . . . , n } ] with respect to the orderedbasis ( , . . . , n ) for π ∈ S n . • R + ( g ) denotes the matrix of its action on V + with respect to the ordered basis(v , v . . . , v n ) for g ∈ G n . • R − ( g ) denotes the matrix of its action on V − with respect to the ordered basis ∪ h ∈ G n \{ e } (cid:16) v h , v h . . . , v n h (cid:17) for g ∈ G n . In this case the ordered basis is the union of ( | G n | −
1) ordered basis indexed byelements of G n \ { e } . Lemma 4.5.
For any ( g , g , . . . , g n ; π ) ∈ G n , the matrices R + (( g , g , . . . , g n ; π )) and ρ def ( π ) are the same. Moreover the eigenvalues of b P ( R + ) are given by Eigenvalues: 1 1 − n n − Proof.
Following the definition of R + and ρ def we have, R + (( g , g , . . . , g n ; π ))(v i ) = R (( g , g , . . . , g n ; π ))(v i ) = v π (i) and ρ def ( π )( i ) = π ( i ) , for 1 ≤ i ≤ n . Thus the first part of the lemma follows. Also the second part of this theoremfollows from Lemma 4.1 and the fact b P ( R + ) = 1 n n − X u =1 ρ def (( u, n )) + ρ def (id) ! = 1 n n X u =1 ρ def (( u, n )) + ρ def (id) ! . (cid:3) Lemma 4.6.
The eigenvalues of b P ( R − ) are given by Eigenvalues: 1 − n n − | G n | −
1) ( | G n | − Proof.
Let I ∗ denote the identity matrix of order ( | G n | − × ( | G n | − h ( n ) := (e , . . . , e , h ; id) for h ∈ G n and( h − ) ( k ) h ( n ) ( k, n ) = (e , . . . , e , h − ↑ , e , . . . , e , h ; ( k, n )) for h ∈ G n .k th position.For g ∈ G n \ { e } and 1 ≤ k < n, we have the following X h ∈ G n R − (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) (v i g ) = | G n | v i g if i = k, n i = k i = n and(33) X h ∈ G n R − ( h ( n ) )(v i g ) = | G n | v i g if i = n i = n. Therefore (33) implies1 | G n | X h ∈ G n R − (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) = I ∗ ⊗ ( I n − M k − M n ) for all k ∈ [ n − . (34) 1 | G n | X h ∈ G n R − (cid:16) h ( n ) (cid:17) = I ∗ ⊗ ( I n − M n ) . (35)Now using (34), (35) and the definition of b P ( R − ) we have, b P ( R − ) = 1 n n − X k =1 | G n | X h ∈ G n R − (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) + 1 | G n | X h ∈ G n R − (cid:16) h ( n ) (cid:17) = 1 n n − X k =1 I ∗ ⊗ ( I n − M k − M n ) + I ∗ ⊗ ( I n − M n ) ! = 1 n I ∗ ⊗ n ( I n − M n ) − n − X k =1 M k ! = (cid:18) − n (cid:19) I ∗ ⊗ ( I n − M n ) . (36)Thus (36) implies that 0 is an eigenvalue of b P ( R − ) with multiplicity ( | G n | −
1) and 1 − n isan eigenvalue of b P ( R − ) with multiplicity ( | G n | − n − (cid:3) Proposition 4.7.
Recall that E k ( X ) is the expectation of X with respect to the probabilitymeasure P ∗ k on G n . Then E k ( X ) = 1 + (( n − | G n | − (cid:16) − n (cid:17) k . Proof.
In the proof we use the fact that the trace of k th power of a matrix is the sum of the k th powers of its eigenvalues. We also know that X ( g ) = trace ( R + ( g )) + trace ( R − ( g )) forall g ∈ G n . Thus from the definition of expectation we have E k ( X ) = X g ∈G n P ∗ k ( g ) (cid:16) trace (cid:16) R + ( g ) (cid:17) + trace (cid:16) R − ( g ) (cid:17)(cid:17) = trace X g ∈G n P ∗ k ( g ) R + ( g ) + trace X g ∈G n P ∗ k ( g ) R − ( g ) (37) = trace (cid:16) d P ∗ k ( R + ) (cid:17) + trace (cid:16) d P ∗ k ( R − ) (cid:17) = trace (cid:18)(cid:16) b P ( R + ) (cid:17) k (cid:19) + trace (cid:18)(cid:16) b P ( R − ) (cid:17) k (cid:19) = 1 + (( n − | G n | − (cid:18) − n (cid:19) k . (38)The equality in (37) holds because trace is linear. The equality in (38) follows from Lemmas4.5 and 4.6. (cid:3) UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 25
Our goal now is to find the expectation E k ( X ) of X with respect to the probability measure P ∗ k . For any g ∈ G n , let us first observe the following:( X ( g )) = (cid:16) trace (cid:16) R + ( g ) (cid:17) + trace (cid:16) R − ( g ) (cid:17)(cid:17) = (cid:16) trace (cid:16) R + ( g ) (cid:17)(cid:17) + 2 (cid:16) trace (cid:16) R − ( g ) (cid:17)(cid:17) (cid:16) trace (cid:16) R + ( g ) (cid:17)(cid:17) + (cid:16) trace (cid:16) R − ( g ) (cid:17)(cid:17) = trace (cid:16) R + ( g ) ⊗ R + ( g ) (cid:17) + 2 trace (cid:16) R − ( g ) ⊗ R + ( g ) (cid:17) + trace (cid:16) R − ( g ) ⊗ R − ( g ) (cid:17) . (39)Expression (39) suggests us to define three representations R : G n → GL ( V + ⊗ V + ), R : G n → GL ( V − ⊗ V + ) and R : G n → GL ( V − ⊗ V − ) of G n . Precisely given as follows: R ( g ) = (cid:16) R + ⊗ R + (cid:17) ( g )( v i ⊗ v j ) = R + ( g )( v i ) ⊗ R + ( g )( v j ) for g ∈ G n , v i ∈ V + , v j ∈ V + , R ( g ) = (cid:16) R − ⊗ R + (cid:17) ( g )( v i ⊗ v j ) = R − ( g )( v i ) ⊗ R + ( g )( v j ) for g ∈ G n , v i ∈ V − , v j ∈ V + , R ( g ) = (cid:16) R − ⊗ R − (cid:17) ( g )( v i ⊗ v j ) = R − ( g )( v i ) ⊗ R − ( g )( v j ) for g ∈ G n , v i ∈ V − , v j ∈ V − . Lemma 4.8. E k ( X ) can be expressed as follows trace (cid:18)(cid:16) b P ( R ) (cid:17) k (cid:19) + 2 trace (cid:18)(cid:16) b P ( R ) (cid:17) k (cid:19) + trace (cid:18)(cid:16) b P ( R ) (cid:17) k (cid:19) . Proof.
Using linearity of trace and ( X ( g )) = trace ( R ( g )) + 2 trace ( R ( g )) + trace ( R ( g ))we have E k ( X ) = X g ∈G n P ∗ k ( g ) (trace ( R ( g )) + 2 trace ( R ( g )) + trace ( R ( g )))= trace X g ∈G n P ∗ k ( g ) R ( g ) + 2 trace X g ∈G n P ∗ k ( g ) R ( g ) + trace X g ∈G n P ∗ k ( g ) R ( g ) = trace (cid:16) d P ∗ k ( R ) (cid:17) + 2 trace (cid:16) d P ∗ k ( R ) (cid:17) + trace (cid:16) d P ∗ k ( R ) (cid:17) = trace (cid:18)(cid:16) b P ( R ) (cid:17) k (cid:19) + 2 trace (cid:18)(cid:16) b P ( R ) (cid:17) k (cid:19) + trace (cid:18)(cid:16) b P ( R ) (cid:17) k (cid:19) . (cid:3) Lemma 4.9.
The eigenvalues of b P ( R ) are given as follows: Eigenvalues: 1 1 − n n − n − n Multiplicities: 2 3( n −
2) 3 n − n − n − n + 5 Proof.
Let us recall that R + (( g , g , . . . , g n ; π )) = ρ def ( π ) for all ( g , g , . . . , g n ; π ) ∈ G n and b P ( R ) = X g ∈G n P ( g ) R ( g ) = X g ∈G n P ( g ) (cid:16) R + ( g ) ⊗ R + ( g ) (cid:17) . Therefore we have b P ( R ) = 1 n | G n | n − X k =1 X h ∈ G n R + (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) ⊗ R + (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) + 1 n | G n | X h ∈ G n R + (cid:16) h ( n ) (cid:17) ⊗ R + (cid:16) h ( n ) (cid:17) = 1 n n − X k =1 ρ def (( k, n )) ⊗ ρ def (( k, n )) + ρ def (id) ⊗ ρ def (id) ! . Thus the lemma follows from Lemma 4.3. (cid:3)
Lemma 4.10.
The eigenvalues of b P ( R ) are given as follows: Eigenvalues: 1 − n − n n − | G n | −
1) ( n − n − | G n | −
1) (2 n − | G n | − Proof.
Let us recall that R + (( g , g , . . . , g n ; π )) = ρ def ( π ) for all ( g , g , . . . , g n ; π ) ∈ G n . Nowusing the definition of b P ( R ) we have b P ( R ) = X g ∈G n P ( g ) R ( g ) = X g ∈G n P ( g ) (cid:16) R − ( g ) ⊗ R + ( g ) (cid:17) = 1 n | G n | n − X k =1 X h ∈ G n R − (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) ⊗ R + (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) + 1 n | G n | X h ∈ G n R − (cid:16) h ( n ) (cid:17) ⊗ R + (cid:16) h ( n ) (cid:17) = 1 n n − X k =1 | G n | X h ∈ G n R − (cid:16) ( h − ) ( k ) h ( n ) ( k, n ) (cid:17) ⊗ ρ def (( k, n ))+ 1 n | G n | X h ∈ G n R − (cid:16) h ( n ) (cid:17) ⊗ ρ def (id) . = 1 n n − X k =1 I ∗ ⊗ ( I n − M k − M n ) ⊗ ρ def (( k, n )) + 1 n I ∗ ⊗ ( I n − M n ) ⊗ ρ def (id)(40) = 1 n I ∗ ⊗ n − X k =1 ( I n − M k − M n ) ⊗ ρ def (( k, n )) + ( I n − M n ) ⊗ ρ def (id) ! = 1 n I ∗ ⊗ ( I n − M n ) ⊗ n − X k =1 ρ def (( k, n )) + ρ def (id) ! − n − X k =1 M k ⊗ ρ def (( k, n )) ! (41)The equality in (40) follows from (34) and (35). If Blockdiag ( A , A , . . . , A n ) denote theblock diagonal matrix with i th block A i for all i ∈ [ n − β i given in (4.2), the right hand side of (41) can be written as1 n I ∗ ⊗ Blockdiag ( I n + β , I n + β , . . . , I n + β n − , O n ) . Therefore the lemma follows from Lemma 4.2. (cid:3)
Let us first prove a lemma which will be useful in finding the eigenvalues of b P ( R ). Thislemma is true in general not only for our setting. Lemma 4.11.
Let G be a finite group. Recall that V = C [ G ] is the complex vector spacewith basis G . Also let ( ρ, V ) be the left regular representation of G i.e. ρ ( g ) X h ∈ G C h h X h ∈ G C h gh , g ∈ G, C h ∈ C . UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 27
Then the eigenvalues of | G | X g ∈ G ρ ( g ) ⊗ ρ ( g − ) are in the closed unit disc D := { z ∈ C : | z | ≤ } .Proof. Before proving the lemma let us recall the definition of a stochastic matrix . A realsquare matrix is said to be stochastic if all its entries are from the interval [0 ,
1] and sum ofthe elements in each row is 1.It can be easily seen that ρ ( g ) is a permutation matrix and thus a stochastic matrix foreach g ∈ G . Therefore ρ ( g ) ⊗ ρ ( g − ) is a stochastic matrix for each g ∈ G . It can be easilyseen that the average of stochastic matrices are stochastic. Therefore | G | X g ∈ G ρ ( g ) ⊗ ρ ( g − ) isa stochastic matrix. Hence the lemma follows from Gerˇshgorin disc theorem [11, Theorem6.1.1]. (cid:3) Corollary 4.12.
Let G be a finite group and V ′ be the vector space spanned by the complexlinear combinations of elements of the set { v g | g ∈ G } , where v g = X h ∈ G h − | G | g . Also let L be the representation of G defined by left regular action on V ′ . Then the eigenvalues of | G | X g ∈ G L ( g ) ⊗ L ( g − ) are in the closed unit disc D := { z ∈ C : | z | ≤ } .Proof. Let V = C [ G ] and ρ denote the left regular representation of G . Also note that V ′ ⊗ V ′ is a subspace of V ⊗ V invariant under | G | X g ∈ G ρ ( g ) ⊗ ρ ( g − ). Then the corollaryfollows follows from the fact | G | X g ∈ G L ( g ) ⊗ L ( g − ) = 1 | G | X g ∈ G ρ ( g ) ⊗ ρ ( g − ) (cid:12)(cid:12)(cid:12) V ′ ⊗ V ′ . (cid:3) Lemma 4.13.
Let L be the representation of G n defined by left regular action on the vectorspace spanned by the set { v g : g ∈ G n } , where v g = X h ∈ G n h − | G n | g . If the eigenvalues of | G n | X g ∈ G n L ( g ) ⊗ L ( g − ) are λ i for ≤ i ≤ ( | G n | − , then the eigenvalues of b P ( R ) aregiven in Table 1 . Also we note that | λ i | ≤ for ≤ i ≤ ( | G n | − in this case.Proof. We prove this lemma by splitting V − ⊗ V − into three subspaces which are invariantunder the action of b P ( R ). The subspaces are given as follows. • Let W be the subspace of V − ⊗ V − spanned by the set of vectors ∪ ≤ i Eigenvalues of b P ( R )diagonal matrix with respect to a certain choice for ordering of the basis elements. Our goalis to use that block diagonal decomposition and obtain the eigenvalues of b P ( R ).First let us notice that the subspaces W i of V − ⊗ V − spanned by the set of vectors (cid:16)n v i x ⊗ v n y : x, y ∈ G n \ { e } o ∪ n v n x ⊗ v i y : x, y ∈ G n \ { e } o(cid:17) is also invariant under the action of b P ( R ) for each i ∈ [ n − 1] and W = W ⊕ · · · ⊕ W n − .We now focus on W = W ⊕ · · · ⊕ W n − . For any 1 ≤ i < n , consider the ordered basis B i of W i as follows: In B i first list all elements of the form v i x ⊗ v n y and then list the elementsof the form v n x ⊗ v i y by maintaining the same ordering of the pair ( x, y ). Using this orderedbasis B i , b P ( R ) (cid:12)(cid:12)(cid:12) W i will be of the form(42) 1 n ! ⊗ | G n | X g ∈ G n L ( g ) ⊗ L ( g − ) . Here the matrix | G n | X g ∈ G n L ( g ) ⊗ L ( g − ) is written with respect to a ordered basis followingthe same ordering as that of ( x, y ) in B i . Since (42) is true for all i ∈ [ n − b P ( R ) (cid:12)(cid:12)(cid:12) W are given by: ± λ i n withmultiplicity ( n − 1) each for 1 ≤ i ≤ ( | G n | − .We now focus on the subspace W ′ of V − ⊗ V − . It can be easily seen that b P ( R )(v i g ⊗ v j h ) = (cid:18) − n (cid:19) (v i g ⊗ v j h ) , for g, h ∈ G n \ { e } , ≤ i, j < n and i = j. Therefore b P ( R ) (cid:12)(cid:12)(cid:12) W ′ is the scalar matrix (cid:16) − n (cid:17) I dim( W ′ ) .The eigenvalues and eigenvectors of b P ( R ) (cid:12)(cid:12)(cid:12) W ′′ are given in Table 2 and hence the lemmafollows. In this table g is a fixed non identity element of G n . (cid:3) UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 29 Eigenvalue Eigenvectors corresponding to number of (independent)eigenvalue given in column 1 vectors in column 21 X h ∈ G n ≤ i ≤ n v i h ⊗ v i hx , for x ∈ G n \ { e } . | G n | − nn − X h ∈ G n ≤ i ≤ n − v i h ⊗ v i hx − n X h ∈ G n v n h ⊗ v n hx , | G n | − x ∈ G n \ { e } .0 v n g ⊗ v n gx − v n h ⊗ v n hx , for x ∈ G n \ { g − } ( | G n | − | G n | − h ∈ G n \ { g, x − } . − n v i g ⊗ v i gx − v i h ⊗ v i hx , for x ∈ G n \ { g − } , ( n − | G n | − | G n | − h ∈ G n \ { g, x − } and 1 ≤ i < n. − n v i g ⊗ v i gx − v g ⊗ v gx for x ∈ G n \ { g − } , ( n − | G n | − ≤ i < n . Table 2. Eigenvectors and eigenvalues of b P ( R ) (cid:12)(cid:12)(cid:12) W ′′ Proposition 4.14. Recall that V k ( X ) is the variance of X with respect to the probabilitymeasure P ∗ k on G n . Then V k ( X ) = | G n | + (cid:16) ( n − | G n | − | G n | (cid:17) (cid:18) − n (cid:19) k + (cid:16) − k (cid:17) n k n − n − ( | G n |− X i =1 λ ki + (cid:16) ( n − n − | G n | − + 2( n − n − | G n | − 1) + n − n + 5 (cid:17) (cid:18) − n (cid:19) k − (( n − | G n | − (cid:18) − n (cid:19) k . Where λ i are defined in Lemma 4.13.Proof. This proposition follows from the definition of variance, Proposition 4.7, Lemmas 4.8,4.9, 4.10, 4.13 and straightforward calculations. (cid:3) Proposition 4.15. Let ǫ ∈ (0 , be arbitrary and k n = j (1 − ǫ ) (cid:16) n log n n log( | G n | − (cid:17)k .If | G n | = o ( n δ ) for all δ > , then lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = 1 .Proof. For any positive constant a , by Chebychev’s inequality, we have(43) P ∗ k (cid:18)n π ∈ G n : | X ( π ) − E k ( X ) | ≤ a q V k ( X ) o(cid:19) ≥ − a . Now we choose a positive constant a such that E k ( X ) − a q V k ( X ) > 0. Then by Markov’sinequality and Proposition 4.4, we have U G n (cid:18)n π ∈ G n : X ( π ) ≥ E k ( X ) − a q V k ( X ) o(cid:19) ≤ E U ( X ) E k ( X ) − a q V k ( X )(44) = 1 E k ( X ) − a q V k ( X ) . Now from the definition of total variation distance, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = sup A ⊂G n | P ∗ k ( A ) − U G n ( A ) |≥ P ∗ k (cid:18)n π ∈ G n : | X ( π ) − E k ( X ) | ≤ a q V k ( X ) o(cid:19) − U G n (cid:18)n π ∈ G n : | X ( π ) − E k ( X ) | ≤ a q V k ( X ) o(cid:19) ≥ P ∗ k (cid:18)n π ∈ G n : | X ( π ) − E k ( X ) | ≤ a q V k ( X ) o(cid:19) − U G n (cid:18)n π ∈ G n : X ( π ) ≥ E k ( X ) − a q V k ( X ) o(cid:19) ≥ − a − E k ( X ) − a q V k ( X ) . (45)The inequality (45) follows by using (43) and (44). In particular, if we take a = E k ( X )2 √ V k ( X ) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV ≥ − V k ( X )( E k ( X )) − E k ( X ) . Recall that ‘ ≈ ’ means ‘asymptotic to’ i.e. a n ≈ b n means lim n →∞ a n b n = 1. Thus for all k ≥ E k ( X ) ≈ n − | G n | − e − kn . (47) V k ( X ) ≈ | G n | + (cid:16) ( n − | G n | − | G n | (cid:17) e − kn − ( n − | G n | e − kn (48) + 1 + ( − k n k − ( | G n |− X i =1 λ ki . UTOFF PHENOMENON FOR THE WARP-TRANSPOSE TOP WITH RANDOM SHUFFLE 31 Now for any ǫ ∈ (0 , 1) and k n = j (1 − ǫ ) (cid:16) n log n + n log( | G n | − (cid:17)k , (47) and (48) impliesthat E k n ( X ) = 1 + n ǫ | G n | ǫ (1 + o (1)) . (49) V k n ( X ) = | G n | + n ǫ | G n | ǫ (1 + o (1)) + n ǫ | G n | ǫ o (1) + 1 n (cid:16) | G n | − (cid:17) o (1) . (50)Again | G n | = o ( n δ ) for all δ > n →∞ (cid:16) | G n | + n ǫ | G n | ǫ (1 + o (1)) + n ǫ | G n | ǫ o (1) + n (1 + ( | G n | − ) o (1) (cid:17)(cid:16) n ǫ | G n | ǫ (1 + o (1)) (cid:17) = 0 , andlim n →∞ 21 + n ǫ | G n | ǫ (1 + o (1)) = 0 . Therefore (46), (49) and (50) implies that lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = 1 if | G n | = o ( n δ ) for all δ > (cid:3) Proof of Theorem 1.3. For any ǫ ∈ (0 , k n = j (1 + ǫ ) (cid:16) n log n + n log ( | G n | − (cid:17)k ,we have k n + 1 ≥ (1 + ǫ ) (cid:18) n log n + 12 n log ( | G n | − (cid:19) = ⇒ e − knn ≤ e n n ǫ ( | G n | − ǫ . Therefore Proposition 3.6 implies0 ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ k n − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < e e nn ǫ ( | Gn |− ǫ − + e n n ǫ ( | G n | − ǫ + 2 e e e nn ǫ ( | Gn |− ǫ − + 2 (cid:16) | b G n | − (cid:17) e n n ǫ ( | G n | − ǫ e e nn ǫ ( | Gn |− ǫ + en e e nn ǫ ( | Gn |− ǫ − < e n n ǫ ( | G n | − ǫ + e n n ǫ ( | G n | − ǫ + 4 e n n ǫ ( | G n | − ǫ (51) + 2 (cid:16) | b G n | − (cid:17) e n n ǫ ( | G n | − ǫ e e nn ǫ ( | Gn |− ǫ + 2 e n n ǫ ( | G n | − ǫ . The inequality in (51) follows from the fact that e x − < x for all 0 < x ≤ 1. If | G n | = o ( n δ )for all δ > 0, then the right hand side of (51) converges to zero as n → ∞ . Therefore wehave the following:(52) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∗ ⌊ (1+ ǫ ) ( n log n + n log( | G n |− ) ⌋ − U G n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) TV = 0 , if | G n | = o ( n δ ) for all δ > . 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