The $k$-cycle shuffling with repeated cards

THE k -CYCLE SHUFFLING WITH REPEA TED CARDS JIAHE SHEN Abstra ct. W e in vestigate the k -cycle shuffle on repeated cards, namely on a deck consisting of l identical copies of each of m card types, with total size n = ml . W e establish asymptotic results for the total v ariation mixing of this shuffle, including cutoff and explicit limiting profiles. F or fixed l , we show that the walk exhibits cutoff at time n k log n with window of order n k , and we iden tify the limiting profile in terms of the total v ariation distance b etw een Poisson distributions arising from quotien t fixed-p oint statistics. When l → ∞ with sufficiently slow growth, more precisely when l = o (log n ) , w e pro v e that the cutoff lo cation shifts to n k  log n − 1 2 log l  , again with windo w of order n k , and that the limiting profile is asymptotically Gaussian, arising from a P oisson comparison after normal approximation. The pro of is based on an approximation of the sh uffling measure by an explicitly tractable auxiliary measure, generalizing the k = 2 case from Jain and Sawhney [7]. The representation- theoretic framework underlying the analysis of this auxiliary measure follows from the work of Hough [6] and Nestoridi and Olesker-T aylor [9]. Keyw ords: Card shuffling, Random walk, Symmetric group, Representation theory Mathematics Sub ject Classification (2020): 60J10 (primary); 60B15, 20C30 (secondary) Contents 1. In tro duction 1 2. Appro ximating the shuffling with an explicitly tractable measure 5 3. F rom tractable measure to total v ariation distance 16 References 19 1. Intr oduction 1.1. Main results. Questions ab out how rep eated shuffling randomizes a deck of cards natu- rally lead to the study of random walks on the symmetric group. Roughly sp eaking, the cutoff phenomenon means that conv ergence to randomness happ ens abruptly: the system remains far from equilibrium for a long time, and then b ecomes well mixed within a muc h shorter time windo w. A landmark result of Diaconis–Shahshahani [5] show ed that the random transposition sh uffle exhibits cutoff at time 1 2 n log n , thereby in troducing represen tation-theoretic tec hniques that ha v e since b ecome central in the sub ject. Much later, T eyssier [12] determined the corre- sp onding limiting profile, pro ving that at time 1 2 n (log n + c ) the total v ariation distance con v erges to d T V (P ois(1 + exp( − c )) , P ois(1)) . F or the random k -cycle sh uffle, Berestyc ki, Schramm, and Zeitouni [3] established cutoff at time n k log n when k is fixed. Hough [6] later extended the cutoff result to the regime 2 ≤ k ≤ o ( n/ log n ) by dev eloping refined character estimates for the symmetric group. Building on the same character-estimate framew ork, Nestoridi and Olesker-T a ylor [9], together with T eyssier’s Date : March 31, 2026. The author thanks Evita Nestoridi for helpful discussions, and ac kno wledges the Simons Cen ter for Geometry and Ph ysics for funding the Columbia/Ston y Bro ok Probability Day where these discussions occurred. The author also thanks Iv an Corwin and Roger V an Peski for reading the draft and providing commen ts. The author ackno wledges supp ort from Iv an Corwin’s NSF grant DMS-2246576 and Simons In vestigator grant 929852. 1 2 JIAHE SHEN appro ximation metho d, determined the limiting profile in the same regime: at time n k (log n + c ) , the total v ariation distance con v erges to d T V (P ois(1 + exp( − c )) , P ois(1)) . More recently , Olesk er-T aylor, T eyssier, and Thév enin [10] developed sharp character b ounds in a substan tially more general setting, strong enough to reco v er cutoff for the full range of k and to suggest corresp onding profile results as w ell. Based on the ab o v e results, it is natural to consider the analogous shuffling problem when the dec k contains rep eated cards, namely when there are l indistinguishable copies of each of m card types. This leads to a Mark o v chain on a quotien t of the symmetric group, a viewp oin t emphasized for instance in the bo ok of Diaconis-F ulman [4, Chapter 4]. The goal of the presen t pap er is to determine the cutoff time and limiting profile for the random k -cycle shuffle in this rep eated-card setting, in a form that reco v ers the classical single-deck results when l = 1 . Before stating our main results, w e first in troduce the basic definitions and notation. F or all n ≥ 1 , let S n b e the symmetric group ov er n symbols, A n ⊂ S n b e the alternating group, and A c n ⊂ S n b e the complemen t of the alternating group. Denote b y U S n , U A n , U A c n the uniform distribution o v er S n , A n , A c n , resp ectively . Definition 1.1. Supp ose n = ml where m, l ≥ 1 . Then w e abbreviate ∼ l \ S n for the set of left cosets ( S l × · · · × S l | {z } m times ) \ S n := { ( S m × · · · × S m | {z } l times ) ◦ σ : σ ∈ S n } , where S l × · · · × S l | {z } m times := { σ ∈ S n : σ ( i ) ≡ i (mo d m ) ∀ 1 ≤ i ≤ n } is a subgroup of S n . Denote by ∼ l \ U S n the uniform distribution o v er ∼ l \ S n . No w w e give the definition of a random k -cycle. Our setup in v olv es parit y constrain ts which follo w from Nestoridi and Olesker-T a ylor [9] and Hough [6]: Definition 1.2. Let P n,k b e the random permutation in S n giv en by P n,k := ( ( i 1 . . . i k ) with probability k n ( n − 1) ··· ( n − k +1) otherwise with probability zero . (1.1) Moreo v er, for any t ≥ 1 , let P ∗ t n,k b e the t -fold con v olution of P n,k . If k is even and t is o dd, w e view P ∗ t n,k as a random v ariable in A c n ; otherwise, we view P ∗ t n,k as a random v ariable in A n . Giv en n = ml with l ≥ 2 , the random v ariable ∼ l \ P ∗ t n,k ∈∼ l \ S n is induced from Definition 1.1 in the obvious wa y . It is worth highligh ting that in the ab o v e definition, w e regard P ∗ t n,k as a random v ariable in A n or its complement, but we regard ∼ l \ P ∗ t n,k ∈∼ l \ S n as a random v ariable in the quotient space of S n . W e b egin with the regime in which the multiplicit y parameter l grows with n . In this case, the rep eated-card structure already changes the lo cation of cutoff and leads to a different limiting profile from the classical one. Theorem 1.3 (Theorem 3.1 in the text) . L et n = ml , such that l = ω (1) but l = o (log n ) . Supp ose 2 ≤ k ≤ n c 0 + o (1) , wher e c 0 ∈ [0 , 1) is an absolute c onstant, and t = n k  log n − 1 2 log l + c  , wher e c ∈ R is an absolute c onstant. Then, we have d T V ( ∼ l \ P ∗ t n,k , ∼ l \ U S n ) = 2Φ(exp( − c ) / 2) − 1 + o (1) . Her e, Φ( x ) := 1 √ 2 π R x −∞ exp( − y 2 / 2) dy denotes the cumulative distribution function of the normal distribution N (0 , 1) . THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 3 Theorem 1.3 shows that once the num ber of rep eated copies tends to infinity , the chain mixes earlier than in the classical k -cycle sh uffle: compared with the usual cutoff time n k log n , the cutoff is adv anced by n 2 k log l . The window remains of order n k , but the limiting profile is now Gaussian rather than Poissonian. As far as we know, this is a new phenomenon in the study of random k -cycle shuffles. W e next turn to the fixed-m ultiplicit y regime. In that case, the cutoff lo cation is the same as in the classical setting, but the limiting profile is shifted by the repeated-card structure. Theorem 1.4 (Theorem 3.2 in the text) . L et n = ml , such that l ≥ 2 is fixe d, and m = ω (1) . Supp ose 2 ≤ k ≤ o ( n/ log n ) , and t = n k (log n + c ) , wher e c ∈ R is an absolute c onstant. Then, we have d T V ( ∼ l \ P ∗ t n,k , ∼ l \ U S n ) = d T V (P ois( l + exp( − c )) , P ois( l )) + o (1) . Her e, P ois( · ) denotes the Poisson distribution. W e emphasize that Theorem 1.3 and Theorem 1.4 concern conv ergence to the uniform dis- tribution on the quotient space ∼ l \ S n , rather than to the uniform distribution on A n or on A c n . This is a feature specific to the rep eated-card setting. Nevertheless, when l = 1 , the conclusion of Theorem 1.4 parallels the classical k -cycle result: the cutoff o ccurs at time n k log n with a windo w of order n k , and the limiting profile is exactly d T V (P ois(1 + exp( − c )) , P ois(1)) . T o the b est of our knowledge, Theorem 1.3 and Theorem 1.4 provide the first explicit deter- mination of b oth the cutoff lo cation and the limiting profile for the random k -cycle sh uffle on rep eated cards. In particular, they exhibit a transition from a P oissonian profile to a Gaussian one as the m ultiplicit y parameter l grows. W e no w briefly outline the proof of our main results. A key step is inspired by the w ork of Jain and Sawhney [7, Theorem 1.3], which treats the case k = 2 . More precisely , w e construct another probability measure on the shuffling group that follo ws a completely different sampling pro cedure from the original random k -cycle walk, but is considerably more amenable to analysis. The adv antage of this auxiliary measure is that its quotient fixed-p oin t statistics can be described explicitly , whic h ultimately allows us to extract the cutoff lo cation and limiting profile. Definition 1.5. Suppose t ∈ Z . Set t ′ := t − ⌊ n log n k ⌋ , and γ t := exp( − k t ′ /n ) . Let ν t n,k denote the measure on S n defined b y the follo wing sampling pro cess: first, sample M t ∈ { 0 , 1 , . . . , n } according to the distribution P [ M t = x ] = P (Pois( γ t ) = x ) / P (P ois( γ t ) ≤ n ) , then sample a uniformly random subset S t of size M t in [ n ] . Finally , (1) When k is o dd, sample a uniformly random element of A [ n ] \ S t and view it as an element of A n b y fixing all of the elements in S t . (2) When k is even, and t is odd, sample a uniformly random element of A c [ n ] \ S t and view it as an element of A c n b y fixing all of the elements in S t . (3) When k is even, and t is ev en, sample a uniformly random element of A [ n ] \ S t and view it as an elemen t of A n b y fixing all of the elements in S t . The next theorem sho ws that the original sh uffling measure can b e well approximated by the auxiliary measure ν t n,k , in the sense that their total v ariation distance is small in the regimes relev ant to our main results. Theorem 1.6 (Theorem 2.1 in the text) . F or n = ω (1) and k ≥ 2 , the fol lowing holds. L et t ′ := t − n log n k . L et P ∗ t n,k b e define d as in Definition 1.2, and ν t n,k b e define d as in Definition 1.5. (1) Supp ose that 2 ≤ k ≤ o ( n/ log n ) , and t ′ = cn/k , wher e c ∈ R is an absolute c onstant. Then, we have d T V ( P ∗ t n,k , ν t n,k ) = o (1) . 4 JIAHE SHEN (2) Supp ose that 2 ≤ k ≤ n c 0 + o (1) , wher e c 0 ∈ [0 , 1) is an absolute c onstant, and | t ′ | ≤ n (log log n − ω (1)) 2 k . Then, we have d T V ( P ∗ t n,k , ν t n,k ) ≤ n c 0 − 1+ o (1) . With Theorem 1.6 in hand, it remains to compare the auxiliary measure ν t n,k with the uniform probabilit y measure on the quotient space. W e first reduce this comparison to the law of the set of quotient fixed p oints. By symmetry , conditional on the n um ber of quotient fixed p oin ts, both measures are uniform ov er all configurations with that cardinality , so the problem further reduces to comparing the distribution of the num ber of quotien t fixed p oints. Since it is well known that, under the uniform measure, the num ber of quotient fixed p oin ts conv erges to a Poisson distribution, our task is thereby reduced to comparing the total v ariation distance b et w een tw o P oisson laws with different parameters, whic h leads to the desired limiting profile. A t the same time, following Theorem 1.6, one also obtains an alternative pro of of the limiting profile theorem of Nestoridi and Olesker-T a ylor [9, Theorem B]. 1.2. What’s next? Our results for the rep eated-card setting suggest several directions for further in v estigation. In the present paper, we only treat the regime l = o (log n ) , that is, when the multiplicit y parameter tends to infinity v ery slo wly . The main reason is technical: the Jain– Sa whney approach underlying Theorem 1.6 only giv es effective control in a relativ ely narrow time interv al around n k log n . Nevertheless, the conclusion of Theorem 1.3 may well remain v alid for substantially larger v alues of l . A basic example is the case k = 2 , m = 2 , and l = n/ 2 . In this regime, the model is exactly the simple exclusion pro cess on the complete graph with n vertices and n/ 2 particles. F or this c hain, Lacoin-Leblond [8, Theorem 1.1] show ed that the mixing time is of order n 2 log min  n 2 , √ n  = n 4 log n, and that the cutoff window has order n . This is fully consistent with the prediction of Theo- rem 1.3, which in this case gives the cutoff lo cation n 2  log n − 1 2 log n 2  = n 4 log n + O ( n ) , together with a windo w of order n . The abov e observ ation motiv ates us to formulate the follo wing conjecture. Conjecture 1.7. L et n = ml = ω (1) , wher e we r e quir e m, l ≥ 2 . Then the r andom tr ansp osition shuffle on r ep e ate d c ar ds exhibits cutoff at time n 2 (log n − 1 2 log l ) with window of or der n . It should b e reasonable to replace random transp ositions (resp. 2 -cycle) in the ab o v e conjec- ture b y random k -cycles, where k is not to o large. In this case, the cutoff time would b e replaced b y n k (log n − 1 2 log l ) and the time windo w would b e replaced b y n k . Another natural direction is to replace the random k -cycle shuffle b y more general conjugacy- in v ariant sh uffles on the symmetric group. Instead of taking the driving measure to b e supp orted on a single conjugacy class of k -cycles, one may consider random w alks generated b y an arbitrary conjugacy class, or more generally b y a class function on S n . F rom the represen tation-theoretic p oin t of view, this would require extending the character estimates used in the present pap er b ey ond the k -cycle setting. In this direction, one may hop e to draw on the recen t sharp character b ounds developed b y Olesker-T a ylor, T eyssier, and Thév enin [10]. It would b e very interesting to understand whether the repeated-card phenomena exhibited in the presen t pap er–notably , the shift in the cutoff lo cation and the transition from a P oisson profile to a Gaussian profile p ersist in such a general setting. THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 5 1.3. Notation. Throughout the pap er, d T V ( · , · ) refers to the total v ariation distance, Pois( · ) refers to the Poisson distribution, and Φ( · ) refers to the cumulativ e distribution function of the normal distribution N (0 , 1) . W e will write [ n ] for the set { 1 , 2 , . . . , n } , and 2 [ n ] to b e the set consisting of all subsets of [ n ] . When n = ml , w e write T i,l := { i, i + m, . . . , i + ( l − 1) m } for all 1 ≤ i ≤ m (i.e., all rep eated cards of a single type), so that [ n ] = F m i =1 T i,l . W e use the symbol | · | for the order of a finite set or the absolute v alue of a complex n um ber. W e write f = O ( g ) to mean that | f | ≤ C | g | for some absolute constant C > 0 , and w e write f = o ( g ) to mean that f /g → 0 . Similarly , we write f = ω ( g ) to mean that f /g → + ∞ . Here all asymptotic notation is understo o d in absolute v alue. Moreo v er, all suc h asymptotic statemen ts are taken along the given sequence of positive in teger inputs. In particular, they are not meant uniformly ov er all suc h inputs, but only with resp ect to the particular sequence under consideration. 1.4. Outline of the pap er. In Section 2, w e prov e Theorem 1.6. Based on the approximation pro vided by Theorem 1.6, w e prov e Theorem 1.3 and Theorem 1.4 in Section 3. 2. Appr o xima ting the shuffling with an explicitl y tra ct able measure The purp ose of this section is to prov e the follo wing theorem, whic h gives a detailed formu- lation of Theorem 1.6. Theorem 2.1. L et ( n N ) N ≥ 1 , ( k N ) N ≥ 1 , and ( t N ) N ≥ 1 b e se quenc es of p ositive inte gers, such that lim N →∞ n N = ∞ . F or al l N ≥ 1 , let t ′ N := t N − n N log n N k N . A lso, let P ∗ t N n N ,k N b e define d as in Definition 1.2, and ν t N n N ,k N b e define d as in Definition 1.5. (1) Supp ose that (a) k N ≥ 2 for al l N ≥ 1 , and lim N →∞ k N log n N n N = 0 . (b) t ′ N = cn N /k N , wher e c ∈ R is an absolute c onstant. Then, we have lim N →∞ d T V ( P ∗ t N n N ,k N , ν t N n N ,k N ) = 0 . (2) Supp ose that (a) k N ≥ 2 for al l N ≥ 1 , and lim sup N →∞ log n N k N ≤ c 0 , wher e c 0 ∈ [0 , 1) is an absolute c onstant. (b) lim N →∞ log log n N − | 2 k N t ′ N /n N | = + ∞ . Then, we have lim sup N →∞ log n N d T V ( P ∗ t N n N ,k N , ν t N n N ,k N ) ≤ c 0 − 1 . F or the rest of the section, for notational ease, w e drop the subscripts and simply write n, k for n N , k N . No w, let us recall necessary bac kground from represen tation theory . 2.1. Nonab elian F ourier transform and related notations. Giv en a finite group G , we let b G denote the set of irreducible represen tations. W e define conv olution f 1 , f 2 : G → C to be ( f 1 ∗ f 2 )( z ) := X x ∈ G f 1 ( x ) f 2 ( x − 1 z ) . F or ρ ∈ b G , let d ρ denote the dimension of ρ . The nonab elian F ourier transform for a function f : G → C is the map b f given b y b f ( ρ ) := X g ∈ G f ( g ) ρ ( g ) ∈ Mat d ρ ( C ) , ∀ ρ ∈ b G. Then, for all f 1 , f 2 : G → C and ρ ∈ b G , w e hav e \ f 1 ∗ f 2 ( ρ ) = b f 1 ( ρ ) · b f 2 ( ρ ) . (2.1) 6 JIAHE SHEN In this pap er, w e alwa ys take G = S n . F ollo wing the standard construction of Sp ech t mo dules, w e can iden tify elemen ts in b S n with partitions of n . F rom no w on, w e use λ ⊢ n to denote that λ is a partition of n . Given a partition λ = ( λ 1 , . . . , λ j ) with λ 1 ≥ · · · ≥ λ j , w e let λ ′ denote the conjugate partition and λ ∗ = ( λ 2 , . . . , λ j ) b e the partition of n − λ 1 obtained by truncating λ . Denote by l ( λ ) the num ber of parts of λ , and λ • := (( λ ′ ) ∗ ) ′ , whic h is obtained from λ by deleting its first column. W e will also apply the notation in tro duced in the following definition, whic h follows from [9] and [6]. Definition 2.2. Supp ose λ ⊢ n has diagonal length m (see the top of [6, page 454]). The F r ob enius notation is the expression λ = ( a 1 , . . . , a m | b 1 , . . . , b m ) , where a i := λ i − i + 1 2 , b i := λ ′ i − i + 1 2 , ∀ 1 ≤ i ≤ m. Let L :=  λ ⊢ n : λ 1 ≥ n − log n or l ( λ ) ≥ n − log n  (2.2) b e the set of partitions with long first row or long first column. It is clear that L = L 1 F L 2 , where L 1 :=  λ ⊢ n : λ 1 ≥ n − log n  , L 2 :=  λ ⊢ n : l ( λ ) ≥ n − log n  . Moreo v er, under the conjugation map, the partitions in L 1 ha v e one-to-one corresp ondence with the partitions in L 2 . F or the rest of this section, we usually use the letter r for n − λ 1 . Probabilit y measures on S n will be viewed as real-v alued functions. In our applications, w e will furthermore restrict attention to functions f whic h are class functions, i.e., functions that are constants on conjugacy classes. Lemma 2.3 (Sch ur’s lemma) . Supp ose f : S n → C is a class function. Then, we have b f ( λ ) = P g ∈ G f ( g ) χ λ ( g ) d λ Id d λ , wher e χ λ is the char acter c orr esp onding to λ ⊢ n . In particular, let P n,k b e the probability measure ov er S n as in (1.1). Then, its F ourier transform is given by b P n,k ( λ ) = χ λ ( τ k ) d λ Id d λ , λ ⊢ n. (2.3) where τ k is an arbitrary k -cycle. F ollowing [9], we will also use the abbreviation s λ ( k ) for χ λ ( τ k ) d λ . In particular, by the ho ok length form ula, w e ha v e d λ = d λ ′ . F urthermore, denote b y S λ ′ and S λ the Sp ec h t mo dule of t yp es λ ′ and λ , and let sgn n b e the sign representation. Then, the natural isomorphism S λ ′ ∼ = S λ ⊗ sgn n leads to the fact that χ λ ′ ( k ) = χ λ ( k ) · ( − 1) k − 1 , and therefore s λ ′ ( k ) = s λ ( k ) · ( − 1) k − 1 . 2.2. Character estimates for P ∗ t n,k . F ollowing (2.1) and (2.3), it is clear that d P ∗ t n,k ( λ ) = s λ ( k ) t Id d λ , λ ⊢ n. In the following, we pro vide quantitativ e b ounds that will b e useful later. THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 7 Lemma 2.4 ([6, Theorem 5(a)]) . L et 0 < ϵ < 1 2 , let r = n − λ 1 and supp ose that r + k + 1 < ( 1 2 − ϵ ) n . Then s λ ( k ) = ( n − r − 1) k n k m Y i =2  1 − k n − (1 + r + λ i − i )  m Y i =1  1 − k n − ( r − λ ′ i + i )  − 1 + O ϵ  exp  k  log (1 + ϵ )( k + 1 + r ) n − k + O ϵ  r − 1 2   . (2.4) Her e, the sup erscript k is the fal ling factorial. Mor e over, when r < k , the err or term on the se c ond line is actual ly zer o. F ollowing [6], w e will abbreviate M T (resp. E T ) for the main (resp. error) term in (2.4). As w e shall see, when we use the ab ov e b ound, the main term is usually close to 1 , and the error term is usually close to zero. Lemma 2.5 ([6, Lemma 13]) . Supp ose that k + r + 1 < n/ 2 , wher e r := n − λ 1 . Then, we have M T ≤ exp( − r k /n ) . Prop osition 2.6. Supp ose r := n − λ 1 ∈ [1 , log n ] , and 2 ≤ k ≤ o ( n/ log n ) . Mor e over, supp ose that t ′ := t − n log n k satisfies | t ′ | ≤ n (log log n − ω (1)) 2 k . Then, we have s λ ( k ) t = exp  − r k t n − r ( k + r ) log n 2 n (1 + o (1) + O (1 /k ))  . Pr o of. W e first estimate the main term in (2.4). W rite P 0 := ( n − r − 1) k n k , P 1 := m Y i =2  1 − k n − (1 + r + λ i − i )  , P 2 := m Y i =1  1 − k n − ( r − λ ′ i + i )  − 1 . W e ha v e P 0 = n − r − k n − r · ( n − r ) · · · ( n − r − k + 1) n ( n − 1) · · · ( n − k + 1) = (1 − k /n )  1 + O ( k r ) n 2  k − 1 Y i =0  (1 − r /n )  1 − (1 + o (1)) i 2 n 2  = (1 − r /n ) k (1 − k /n )  1 − (1 + o (1)) k 2 r + O ( k r ) 2 n 2  . (2.5) F ollowing the definition of the F rob enius notation, it is clear that m ( m − 1) ≤ r and P m i =2 λ i ≤ r . Therefore, we hav e P 1 = m Y i =2  (1 − k /n ) ·  1 − k (1 + r + λ i − i ) n 2 − k n 3 − o (1)  = (1 − k /n ) m − 1 ·  1 − k P m i =2 (1 + r + λ i − i ) n 2 − k n 3 − o (1)  = (1 − k /n ) m − 1 ·  1 − k ( m + O (1)) r n 2  . (2.6) Notice that P m i =1 λ ′ i ≤ r + m , we also ha v e P 2 = m Y i =1  (1 − k /n ) − 1 ·  1 + k ( r − λ ′ i + i ) n 2 + k n 3 − o (1)  = (1 − k /n ) − m ·  1 + k P m i =1 ( r − λ ′ i + i ) n 2 + k n 3 − o (1)  = (1 − k /n ) − m ·  1 + k ( m + O (1)) r n 2  . (2.7) 8 JIAHE SHEN Com bining the estimates in (2.5), (2.6) and (2.7), we ha v e M T = P 0 P 1 P 2 = (1 − r /n ) k  1 − (1 + o (1)) k 2 r + O ( k r ) 2 n 2  = exp  − r k n − (1 + o (1))( k 2 r + k r 2 ) + O ( k r ) 2 n 2  . (2.8) W e no w deal with the error term in (2.4). When k > r , the error term is zero. If k ≤ r , then b y taking ϵ = 0 . 1 we ha v e E T = O (exp( k [( − 1 + o (1)) log n ])) = O  n ( − 1+ o (1)) k  = O  1 n 3 − o (1)  (2.9) when k ≥ 3 , and E T = O (exp(2(log r − log n + O (1))) = O  r 2 n 2  (2.10) when k = 2 . Therefore, b y Lemma 2.4 and the estimates in (2.8), (2.9) and (2.10), w e ha v e s λ ( k ) = M T + E T = exp  − r k n − (1 + o (1))( k 2 r + k r 2 ) + O ( k r + r 2 ) 2 n 2  , and therefore s λ ( k ) t = exp  − r k t n − t · (1 + o (1))( k 2 r + k r 2 ) + O ( k r + r 2 ) 2 n 2  = exp  − r k t n − r ( k + r ) log n 2 n (1 + o (1) + O (1 /k ))  (2.11) since t = (1 + o (1)) n log n/k . □ The follo wing result is a technical adjustment of Prop osition 2.6. How ev er, in this case, only an upp er b ound is needed. Prop osition 2.7. L et r := n − λ 1 . Supp ose that one of the fol lowing holds: (1) r ∈ [log n, n 5 / 6 ] , and 2 ≤ k ≤ o ( n/ log n ) . (2) r ∈ [ n 5 / 6 , 0 . 499 n ] , and 6 log n ≤ k ≤ o ( n/ log n ) . Mor e over, supp ose that t ′ := t − n log n k satisfies | t ′ | ≤ n (log log n − ω (1)) 2 k . Then, we have | s λ ( k ) | 2 t ≤ exp ( − 2 r log n + r log r − ω ( r )) . Pr o of. F ollowin g Lemma 2.5, w e hav e M T ≤ exp( − r k /n ) . (2.12) No w, let us b ound the error term by taking ϵ = 10 − 10 . F or case (1), when k > r , the error term is zero. If k ≤ r , w e hav e that E T = O (exp( k [( − 1 + o (1)) log n r ])) = O   r n  (1 − o (1)) k  , (2.13) and therefore | s λ ( k ) | = M T + E T ≤ exp( − r k /n ) + O   r n  (1 − o (1)) k  = exp  − r k n + O   r n  (1 − o (1)) k  . Notice that 2 k t/n ≥ 2 log n − log log n + ω (1) , log r ≥ log log n, w e ha v e − 2 r k t/n ≤ − 2 r log n + 2 3 r log r − ω ( r ) . Moreov er, w e hav e t ·  r n  (1 − o (1)) k ≤ n log n ·  r n  2 − o (1) = o ( r ) , and thus | s λ ( k ) | 2 t ≤ exp  − 2 r k t n + O  t ·  r n  (1 − o (1)) k  ≤ exp ( − 2 r log n + r log r − ω ( r )) . THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 9 F or case (2), when k > r , the error term is zero. If k ≤ r , we hav e E T = O  exp  k · log 1 2  = O (exp( − k log 2)) , (2.14) and therefore | s λ ( k ) | = M T + E T ≤ exp( − r k /n ) + O (exp( − k log 2)) = exp  − r k n  (1 + O (exp( − k (log 2 − 0 . 5))) = exp  − r k n + O ( n − 1 . 15 )  . (2.15) Here, in the last line, w e are using the facts that k ≥ 6 log n and 6(log 2 − 0 . 5) ≥ 1 . 15 . F or the same reason as in case (1), w e hav e − 2 r k t/n ≤ − 2 r log n + r log r − ω ( r ) , and therefore | s λ ( k ) | 2 t ≤ exp  − 2 r k t n + O  tn − 1 . 15   ≤ exp ( − 2 r log n + r log r − ω ( r )) . Our b ounds for case (1) and (2) together giv e the pro of. □ Prop osition 2.8. Supp ose that r := n − λ 1 ∈ [ n 5 / 6 , 0 . 499 n ] . F urthermor e, supp ose 2 ≤ k ≤ 6 log n . Mor e over, supp ose that t ′ := t − n log n k satisfies | t ′ | ≤ n (log log n − ω (1)) 2 k . Then, for sufficiently lar ge n , we have | s λ ( k ) | 2 t ≤ exp( − 2 r log n + (2 / 3 + o (1)) r log r ) . When the parameters r , k lie in the range of Prop osition 2.8, the b ound in Lemma 2.4 no longer w orks. Therefore, in order to prov e Prop osition 2.8, we need some new estimates, which w e list b elo w. Lemma 2.9 ([6, Lemma 14]) . Supp ose that r := n − λ 1 ∈ [ n 5 / 6 , 0 . 499 n ] . F urthermor e, supp ose 2 ≤ k ≤ 6 log n . Then, we have the b ound | s λ ( k ) | ≤  1 + O  log n n 1 / 4    X a i >kn 1 / 2 a k i n k + X b i >kn 1 / 2 b k i n k   + O  exp( − k ) log n n 1 / 4  . Lemma 2.10 ([6, Lemma 15]) . L et k ≥ 2 , and let r := n − λ 1 ∈ [ n 5 / 6 , 0 . 499 n ] . Assuming l ( λ ) ≤ λ 1 . Then, we have X a i >kn 1 / 2 a k i n k + X b i >kn 1 / 2 b k i n k ≤ exp  − r k 2 n  . Mor e over, ther e exists a c onstant c 0 > 0 such that the fol lowing holds. Supp ose r > c 0 n , then X a i >kn 1 / 2 a k i n k + X b i >kn 1 / 2 b k i n k ≤ exp  − r k n + k r 2 n 2  . Pr o of of Pr op osition 2.8. Let c 0 b e the same as in Lemma 2.10. On the one hand, if r ≤ min { c 0 , 1 / 5 } n , w e hav e r 2 n 2 ≤ r 5 n ≤ r log( n 5 / 6 ) 4 n log n ≤ r log r 4 n log n . Therefore, log   X a i >kn 1 / 2 a k i n k + X b i >kn 1 / 2 b k i n k   ≤ exp  − r k n + k r 2 n 2  ≤ exp  − r k n + k r log r 4 n log n  . 10 JIAHE SHEN In this case, applying Lemma 2.9, w e hav e | s λ ( k ) | ≤  1 + O  log n n 1 / 4  exp  − r k n + k r log r 4 n log n  + O  exp( − k ) log n n 1 / 4  ≤ exp  − r k n + r k log r 3 n log n  . (2.16) Here, the last line follows b ecause r ≥ n 5 / 6 so that r log r n log n ≥ n − 1 / 6+ o (1) . On the other hand, if r ≥ min { c 0 , 1 / 5 } n , w e already hav e − rk 2 n ≤ − rk n + rk log r 4 n log n when n is sufficiently large, and therefore the inequality | s λ ( k ) | ≤ exp  − rk n + rk log r 3 n log n  follo ws the same w a y as the ab o v e. Thus, when n is sufficien tly large, the inequalit y | s λ ( k ) | 2 t ≤ exp  − 2 r k t n + 2 r log r k t 3 n log n  = exp( − 2 r log n + (2 / 3 + o (1)) r log r ) alw a ys holds. □ The following result is quoted from [9, Theorem 3.5]. Prop osition 2.11. Supp ose that 6 log n < k = o ( n/ log n ) , and let θ := 0 . 68 , so that exp( − θ ) > 0 . 506 . Consider λ ⊢ n with b 1 ≤ a 1 < e − θ n . Then, we have | s λ ( k ) | ≤ exp( − 0 . 51 k ) . Pr o of. Cho ose θ ′ := 0 . 677 , so that θ ′ − 1 6 > 0 . 51 . No w, insp ection of the pro of of [6, Theorem 5(b)] gives the b ound | s λ ( k ) | ≤ exp( k ( − θ ′ + 1 / 6 + o (1))) ≤ exp( − 0 . 51 k ) . □ Remark 2.12. In the original version, [6, Theorem 5(b)] pro vides the slightly w eak er b ound | s λ ( k ) | ≤ exp( − 0 . 5 k ) . F or our tec hnical purposes this estimate is not sufficien t (see the pro of of Theorem 2.15), since we need a strict constan t improv emen t in the exp onent (the improv emen t ma y b e arbitrarily small, but must b e p ositive). F ortunately , by follo wing the pro of of [6, Theorem 5(b)] and optimizing the numerical constants, one can obtain such an impro v emen t with no additional ideas. Prop osition 2.13 ([9, Lemma 3.6]) . The fol lowing statement holds when n is sufficiently lar ge. Assume that 2 ≤ k ≤ 6 log n . L et λ ⊢ n such that l ( λ ) ≤ λ 1 and r := n − λ 1 ∈ [ 1 3 n, n ] . Then, we have | s λ ( k ) | ≤ exp  − 3 r k 5 n  . Lemma 2.14 ([7, Lemma 3.3]) . L et λ ⊢ n , and r := n − λ 1 . Then, we have (1) d λ ≤  n r  · √ r ! ≤ n r / √ r ! . (2) When r ≤ log n , we have d λ =  n r  · d λ ∗  1 − r n + O ( n − 2+ o (1) )  . The following b ound generalizes [7, Lemma 3.4]. It shows that the sum P λ ⊢ n d 2 λ | s λ ( k ) | 2 t is mainly contributed by partitions in L . Theorem 2.15. Supp ose that 2 ≤ k ≤ o ( n/ log n ) . Mor e over, supp ose that t ′ := t − n log n k satisfies | t ′ | ≤ n (log log n − ω (1)) 2 k . Then, we have X λ ⊢ n λ / ∈L d 2 λ | s λ ( k ) | 2 t = n − ω (1) . THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 11 Pr o of. Step 1: reduce to l ( λ ) ≤ λ 1 . Partition conjugation preserves dimensions ( d λ ′ = d λ ), and since the step measure is a class function supp orted on k -cycles we hav e | s λ ′ ( k ) | = | s λ ( k ) | . Moreo v er, λ / ∈ L implies λ ′ / ∈ L b y symmetry of (2.2). Hence it suffices to bound X λ ⊢ n λ / ∈L l ( λ ) ≤ λ 1 d 2 λ | s λ ( k ) | 2 t , (2.17) and the full sum is at most twice (2.17). Step 2: split b y the size of r = n − λ 1 . Fix λ with l ( λ ) ≤ λ 1 and λ / ∈ L , and write r := n − λ 1 . Then, we hav e r ≥ log n . Denote by p ( r ) the partition num ber of r . Recall that the Hardy-Raman ujan form ula for partitions gives the asymptotic estimate p ( r ) ≲ exp( π p 2 r / 3) . Moreo v er, we ha v e the estimate d 2 λ ≤ n 2 r r ! giv en in Lemma 2.14. W e will use these b ounds in the follo wing three regimes. Regime I: r ∈ [log n, n 5 / 6 ] . Applying Prop osition 2.7, we hav e | s λ ( k ) | 2 t ≤ exp ( − 2 r log n + r log r − ω ( r )) , and therefore X λ 1 = n − r d 2 λ | s λ ( k ) | 2 t ≤ n 2 r r ! · p ( r ) · | s λ ( k ) | 2 t ≤ exp( − ω ( r )) = n − ω (1) . Summing ov er r giv es n − ω (1) . Regime I I: r ∈ [ n 5 / 6 , 0 . 499 n ] . W e split further according to k . (1) Supp ose 2 ≤ k ≤ 6 log n . Applying Prop osition 2.8, w e ha v e | s λ ( k ) | 2 t ≤ exp( − 2 r log n + (2 / 3 + o (1)) r log r ) , and therefore X λ 1 = n − r d 2 λ | s λ ( k ) | 2 t ≤ n 2 r r ! · p ( r ) · | s λ ( k ) | 2 t ≤ exp((1 / 3 − o (1)) r log r ) = n − ω (1) . Summing ov er r giv es n − ω (1) . (2) Supp ose 6 log n < k ≤ o ( n/ log n ) . Applying Prop osition 2.7, w e hav e | s λ ( k ) | 2 t ≤ exp ( − 2 r log n + r log r − ω ( r )) , and therefore X λ 1 = n − r d 2 λ | s λ ( k ) | 2 t ≤ n 2 r r ! · p ( r ) · | s λ ( k ) | 2 t ≤ exp( − ω ( r )) = n − ω (1) . Summing ov er r giv es n − ω (1) . Regime I I I: r ∈ [0 . 499 n, n ] . W e split further according to k . (1) Supp ose 2 ≤ k ≤ 6 log n . Applying Prop osition 2.13, w e ha v e | s λ ( k ) | 2 t ≤ exp  − 6 r k t 5 n  ≤ exp( − 1 . 1 r log n ) when n is sufficien tly large, and therefore X λ 1 = n − r d 2 λ | s λ ( k ) | 2 t ≤ n 2 r r ! · p ( r ) · exp( − 1 . 1 r log n ) = exp(0 . 1 r log r + O ( r )) = n − ω (1) . Summing ov er r giv es n − ω (1) . 12 JIAHE SHEN (2) Supp ose 6 log n < k ≤ o ( n/ log n ) . Applying Prop osition 2.11, w e hav e | s λ ( k ) | 2 t ≤ exp( − 1 . 02 k t ) ≤ exp( − 1 . 01 n log n ) when n is sufficien tly large, and therefore X λ 1 = n − r d 2 λ | s λ ( k ) | 2 t ≤  X λ ⊢ n d 2 λ  exp( − 1 . 01 n log n ) = n ! exp( − 1 . 01 n log n ) = n − ω (1) . Summing ov er r giv es n − ω (1) . Com bining the three regimes discussed ab o v e, we complete the proof. □ 2.3. Represen tation-theoretic preliminaries for the F ourier transform of ν t n,k . Fix in tegers n ≥ 1 and 0 ≤ M ≤ n/ 3 . In this case, for λ ⊢ n , at most one of the follo wing holds: (1) λ 1 ≥ n − M . (2) l ( λ ) ≥ n − M . Define a probability measure ξ M (resp. ξ c M ) on A n as follo ws: choose a subset S ⊂ [ n ] uniformly among all subsets of size M , then sample a p erm utation π uniformly from A [ n ] \ S (resp. A c [ n ] \ S ), and extend it to an elemen t of A n (resp. A c n ) by fixing every p oin t of S . The follo wing lemma gives an expression for the F ourier transform of ξ M and ξ c M . Theorem 2.16 (F ourier transform of ξ M and ξ c M ) . W e have c ξ M ( λ ) = K λ,µ + K λ ′ ,µ d λ Id d λ , c ξ c M ( λ ) = K λ,µ − K λ ′ ,µ d λ Id d λ , wher e µ = µ M := ( n − M , 1 , 1 , . . . , 1) ⊢ n, and K α,µ denotes the Kostka numb er. Mor e over, if λ 1 < n − M and l ( λ ) < n − M , then c ξ M ( λ ) = c ξ c M ( λ ) = 0 . If λ 1 ≥ n − M , we have c ξ M ( λ ) =  M n − λ 1  d λ ∗ d λ Id d λ , c ξ c M ( λ ) =  M n − λ 1  d λ ∗ d λ Id d λ . If l ( λ ) ≥ n − M , we have c ξ M ( λ ) =  M n − l ( λ )  d λ • d λ Id d λ , c ξ c M ( λ ) = −  M n − l ( λ )  d λ • d λ Id d λ . Pr o of. W e will only pro v e the theorem for ξ M , since the pro of for ξ c M is analogous. Step 1: rewrite as a subgroup av erage. Denote H := S { 1 } × · · · × S { M } × A { M +1 ,...,n } . By conjugation-in v ariance of χ λ , w e may write c ξ M ( λ ) = a λ Id d λ , where a λ = 1 d λ · X σ ∈ S n ξ M ( σ ) χ λ ( σ ) = 1 d λ · 1 | A n − M | X σ ∈ H χ λ ( σ ) . Let 1 H denote the trivial c haracter of H . By F rob enius recipro city , w e hav e 1 | A n − M | X σ ∈ H χ λ ( σ ) = D 1 H , Res S n H χ λ E H = D Ind S n H 1 H , χ λ E S n . Therefore, we hav e a λ = 1 d λ D Ind S n H 1 H , χ λ E S n . Th us, it remains to identify the coefficient of χ λ when we write Ind S n H 1 H as a linear sum of the form χ α with α ⊢ n , which is what we will do in the follo wing. THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 13 Step 2: split Ind S n H 1 H in to t w o induced pieces. Let S µ := S { 1 } × · · · × S { M } × S { M +1 ,...,n } , the Y oung subgroup corresp onding to µ = ( n − M , 1 M ) . Then H ⊂ S µ is a subgroup of index 2 , so the induced representation satisfies Ind S µ H 1 H ∼ = 1 S µ ⊕ g sgn , where g sgn is the inflation of the sign represen tation from the factor S { M +1 ,...,n } to S µ . By transitivit y of induction, Ind S n H 1 H ∼ = Ind S n S µ 1 S µ ⊕ Ind S n S µ g sgn . (2.18) Step 3: identify multiplicities using Y oung’s rule and tensoring by the sign rep- resen tation. Recall that Y oung’s rule gives the decomp osition Ind S n S µ 1 S µ = X α ⊢ n K α,µ χ α , so the multiplicit y of χ λ in the first summand of (2.18) is K λ,µ . Next, observe that g sgn = Res S n S µ sgn n (since the S { i } factors are trivial for all 1 ≤ i ≤ M ), hence Ind S n S µ g sgn = Ind S n S µ  Res S n S µ sgn n  ∼ =  Ind S n S µ 1 S µ  ⊗ sgn n . Notice that tensoring an irreducible S n -represen tation with sgn n conjugates the partition, i.e., χ α ⊗ sgn n = χ α ′ . Therefore, Ind S n S µ g sgn = X α ⊢ n K α,µ χ α ′ , so the m ultiplicit y of χ λ in the second summand of (2.18) is K λ ′ ,µ . Com bining the tw o summands yields D Ind S n H 1 H , χ λ E S n = K λ,µ + K λ ′ ,µ , and hence c ξ M ( λ ) = K λ,µ + K λ ′ ,µ d λ Id d λ . Step 4: explicit form of the K ostk a num ber. As in the pro of of [7, Lemma 3.5], we hav e K λ,µ = ( 0 λ 1 < n − M d λ ∗  M n − λ 1  λ 1 ≥ n − M . Similarly , w e hav e the conjugation analogue given by K λ ′ ,µ = ( 0 l ( λ ) < n − M  M n − l ( λ )  d λ • l ( λ ) ≥ n − M . Substituting into the Kostk a-num ber form ula giv es the claimed explicit expression. □ 2.4. Pro of of Theorem 2.1. W e already hav e the necessary preparation to pro v e Theorem 2.1. Pr o of of The or em 2.1. F or the rest of the pro of, unless otherwise sp ecified, the deductions w ork for b oth cases (1) and (2). In order to simplify computations, w e consider the follo wing truncated v ersion µ t n,k of ν t n,k , whic h is defined in a similar fashion, except that the distribution of M t (i.e., the size of the random subset S t ) is given by P [ M t = x ] = P [P ois( γ t ) = x ] / P [P ois( γ t ) ≤ log n ] , x ≤ log n. Here, γ t := exp( − k t ′ /n ) follows from Definition 1.5. Due to our assumption on t ′ for b oth cases (1) and (2), w e hav e γ t = o ( √ log n ) , and therefore P [P ois( γ t ) ≥ log n ] ≤ P [Pois( p log n ) ≥ log n ] = n − ω (1) . Therefore, it follo ws from the natural decoupling that d T V ( ν t n,k , µ t n,k ) = n − ω (1) . Th us, it suffices to prov e the same b ound for d T V ( P ∗ t n,k , µ t n,k ) . Let f 1 b e the probability densit y function of P ∗ t n,k , 14 JIAHE SHEN and f 2 b e the probabilit y densit y function of µ t n,k . Using the Cauc h y–Sc h w arz inequalit y follow ed b y the Planc herel form ula, w e ha v e that (the dagger sup erscript refers to the conjugate transp ose matrix) 4 · d T V ( P ∗ t n,k , µ t n,k ) 2 ≤ n ! X σ ∈ S n ( f 1 ( σ ) − f 2 ( σ )) 2 = X λ ⊢ n d λ T r  ( b f 1 ( λ ) − b f 2 ( λ ))( b f 1 ( λ ) − b f 2 ( λ )) †  = X λ / ∈L d 2 λ | s λ ( k ) | 2 t + X λ ∈L d λ T r  ( b f 1 ( λ ) − b f 2 ( λ ))( b f 1 ( λ ) − b f 2 ( λ )) †  = n − ω (1) + X λ ∈L d λ T r  ( b f 1 ( λ ) − b f 2 ( λ ))( b f 1 ( λ ) − b f 2 ( λ )) †  . (2.19) Here, the third line follo ws from Theorem 2.16 since b f 2 ( λ ) = 0 when λ / ∈ L , and the final line follo ws from Theorem 2.15. Moreo v er, due to the symmetry under conjugation, the partitions in L 1 and L 2 con tribute the same to the whole sum, and we only need to pro v e X λ ∈L 1 d λ T r  ( b f 1 ( λ ) − b f 2 ( λ ))( b f 1 ( λ ) − b f 2 ( λ )) †  = ( o (1) Case (1) n 2 c 0 − 2 − o (1) Case (2) . No w, supp ose that λ ∈ L 1 , and denote r := n − λ 1 . On the one hand, by Prop osition 2.6, w e ha v e b f 1 ( λ ) = s λ ( k ) t Id d λ = exp  − r k t n − r ( k + r ) log n 2 n (1 + o (1) + O (1 /k ))  Id d λ . (2.20) On the other hand, notice that µ t n,k ∼ ( P log n l =0 ξ c l · P [Pois( γ t )= l ] P [Pois( γ t ) ≤ log n ] t odd , k even P log n l =0 ξ l · P [Pois( γ t )= l ] P [Pois( γ t ) ≤ log n ] else , where ξ l , ξ c l are the distributions in the statement of Theorem 2.16. Therefore, it follows from Theorem 2.16 and Lemma 2.14 that b f 2 ( λ ) = P [P ois( γ t ) ≤ log n ] − 1 · log n X l =0 d λ ∗  l r  d λ P [P ois( γ t ) = l ] Id d λ =  1 − r n + O ( n − 2+ o (1) )  log n X l =0  l r  · r ! n − r · γ l t e − γ t l ! Id d λ =  1 − r n + O ( n − 2+ o (1) )  log n X l = r n − r · γ l t e − γ t ( l − r )! Id d λ =  1 − r n + O ( n − 2+ o (1) )  n − r · γ r t · P [Pois( γ t ) ≤ log n − r ] Id d λ =  1 − r n + O ( n − 2+ o (1) )  exp ( − r k t/n ) P [P ois( γ t ) ≤ log n − r ] Id d λ . (2.21) W e are close to done at this p oin t. Supp ose b f 1 ( λ ) − b f 2 ( λ ) = α λ Id d λ , λ ∈ L 1 , so that X λ ∈L 1 d λ T r  ( b f 1 ( λ ) − b f 2 ( λ ))( b f 1 ( λ ) − b f 2 ( λ )) †  = X λ ∈L 1 d 2 λ | α λ | 2 . W e break the ab ov e sum in to tw o parts, dep ending on whether r ≥ 1 2 log n or not. (1) Supp ose r ≥ 1 2 log n . In this case, it is clear that r ( k + r ) log n 2 n (1 + o (1) + O (1 /k )) ≥ o (1) THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 15 for b oth cases (1) and (2). This implies b f 1 ( λ ) ≤ exp( − r k t/n )(1 + o (1)) . Therefore, we can use the rough b ound b f 1 ( λ ) − b f 2 ( λ ) = α λ Id d λ where | α λ | ≤ 3 exp( − r k t/n ) , along with the dimension b ound d 2 λ ≤ n 2 r /r ! in Lemma 2.14 to see that X n − λ 1 ∈ [ 1 2 log n, log n ] d 2 λ | α λ | 2 ≤ 9 X r ∈ [ 1 2 log n, log n ] X λ 1 = n − r exp( − 2 r k t/n ) n 2 r r ! ≤ O (log n ) · max r ∈ [ 1 2 log n, log n ] exp( − 2 r k t ′ /n ) r ! exp  π p 2 r / 3  = n − ω (1) . (2.22) Here, in the last inequalit y , we used Stirling’s form ula and the assumption | 2 kt ′ /n | = log log n − ω (1) for b oth cases (1) and (2). (2) Supp ose r < 1 2 log n . In this case, we hav e P [P ois( γ t ) ≤ log n − r ] ≥ P  P ois( p log n ) ≤ 1 2 log n  ≥ 1 − n − ω (1) , and therefore b f 2 ( λ ) =  1 − r n + O ( n − 2+ o (1) )  exp ( − r k t/n ) . Also, by Lemma 2.14, w e hav e X λ 1 = n − r d 2 λ ≤  n r  2 X µ ⊢ r d 2 µ =  n r  2 · r ! ≤ n 2 r r ! . (2.23) F or case (1), we split the interv al [1 , 1 2 log n ) in to tw o smaller parts. If r < min { q n k log n , 1 2 log n } , w e hav e r ( k + r ) log n 2 n (1 + o (1) + O (1 /k )) = o (1) , and therefore α λ = exp( − r k t/n )  exp  − r ( k + r ) log n 2 n (1 + o (1) + O (1 /k ))  −  1 − r n + O ( n − 2+ o (1) )   = O  r ( k + r ) log n 2 n  exp( − r k t/n ) . If q n k log n ≤ r < 1 2 log n , we hav e k = ω (1) , whic h implies r ( k + r ) log n 2 n (1 + o (1) + O (1 /k )) = r ( k + r ) log n 2 n (1 + o (1)) , and therefore α λ = exp( − r k t/n )  exp  − r ( k + r ) log n 2 n (1 + o (1) + O (1 /k ))  −  1 − r n + O ( n − 2+ o (1) )   = O (exp( − r k t/n )) . 16 JIAHE SHEN Summing up the ab o v e tw o cases and applying (2.23), we ha v e X n − λ 1 < 1 2 log n d 2 λ | α λ | 2 ≤ X n − λ 1 < q n k log n d 2 λ | α λ | 2 + X q n k log n ≤ n − λ 1 < 1 2 log n d 2 λ | α λ | 2 = X r< q n k log n O  r ( k + r ) log n 2 n  2 n 2 r r ! exp( − 2 r k t/n ) + X q n k log n ≤ r< 1 2 log n n 2 r r ! O (exp( − 2 r k t/n )) ≤ X r ≥ 0 O  r ( k + r ) log n 2 n  2 exp( − 2 cr ) r ! + X r ≥ q n k log n O  exp( − 2 cr ) r !  = O  k 2 (log n ) 2 n 2  + o (1) = o (1) . (2.24) F or case (2), we hav e r ( k + r ) log n 2 n (1 + o (1) + O (1 /k )) = O ( n c 0 − 1+ o (1) ) , and therefore α λ = exp( − r k t/n )  exp( O ( n c 0 − 1+ o (1) )) −  1 − r n + O ( n − 2+ o (1) )  ≤ n c 0 − 1+ o (1) exp( − r k t/n ) . Applying (2.23), w e hav e X n − λ 1 < 1 2 log n d 2 λ | α λ | 2 ≤ n 2 c 0 − 2+ o (1) X r< 1 2 log n n 2 r r ! exp( − 2 r k t/n ) ≤ n 2 c 0 − 2+ o (1) X r ≥ 0 exp( − 2 r k t ′ /n ) r ! = n 2 c 0 − 2+ o (1) . (2.25) Here, we use the fact that | 2 k t ′ /n | = log log n − ω (1) , which implies exp exp( − 2 k t ′ /n ) = n o (1) . Com bining our discussion of r ∈ [1 , 1 2 log n ) and r ∈ [ 1 2 log n, log n ] ab o v e, w e complete the pro of. □ 3. Fr om tra ct able measure to tot al v aria tion dist ance In this section, w e pro v e the following theorems, which elab orate on Theorem 1.3 and The- orem 1.4 in detail. Theorem 3.1. L et ( n N ) N ≥ 1 , ( m N ) N ≥ 1 , ( l N ) N ≥ 1 , ( k N ) N ≥ 1 , and ( t N ) N ≥ 1 b e se quenc es of p ositive inte gers that satisfy the fol lowing: (1) n N = m N · l N for al l N ≥ 1 . (2) lim N →∞ l N = ∞ , but lim N →∞ l N log n N = 0 . (3) k N ≥ 2 for al l N ≥ 1 , and lim sup N →∞ log n N k N ≤ c 0 , wher e c 0 ∈ [0 , 1) is an absolute c onstant. (4) t N = n N k N (log n N − 1 2 log l N + c ) for al l N ≥ 1 , wher e c ∈ R is an absolute c onstant. THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 17 Then, we have lim N →∞ d T V ( ∼ l N \ P ∗ t N n N ,k N , ∼ l N \ U S n N ) = 2Φ(exp( − c ) / 2) − 1 . Theorem 3.2. L et l ≥ 2 b e a fixe d inte ger, ( n N ) N ≥ 1 , ( m N ) N ≥ 1 , ( k N ) N ≥ 1 , and ( t N ) N ≥ 1 b e se quenc es of p ositive inte gers that satisfy the fol lowing: (1) n N = m N · l for al l N ≥ 1 . (2) lim N →∞ m N = ∞ . (3) k N ≥ 2 for al l N ≥ 1 , and lim N →∞ k N log n N n N = 0 . (4) t N = n N k N (log n N + c ) for al l N ≥ 1 , wher e c ∈ R is an absolute c onstant. Then, we have lim N →∞ d T V ( ∼ l \ P ∗ t N n N ,k N , ∼ l \ U S n N ) = d T V (P ois( l + exp( − c )) , P ois( l )) . As in Section 2, for notational conv enience, we suppress the subscript N for the rest of this section. Definition 3.3. Let n = ml . W e define the fixe d p oint set F n,l : S n → 2 [ n ] as follo ws. F or σ ∈ S n , let F n,l ( σ ) := { i ∈ [ n ] : σ ( i ) ≡ i (mod m ) } . The function F n,l naturally descends to a well-defined function on ∼ l \ S n , which we still denote b y F n,l . W e sa y that an element σ ∈ S n is clus ter e d if | F n,l ( σ ) ∩ T i,l | ≥ 2 for some 1 ≤ i ≤ m . Definition 3.4. Retain the notation and the sampling pro cedure from Definition 1.5. W e define a probability measure ν sym ,t n,k on S n b y following exactly the same construction as for ν t n,k , except that in the final step, after sampling S t , we c hoose a uniformly random element of S [ n ] \ S t and then view it as an elemen t of S n b y fixing ev ery elemen t of S t . W e denote by ν sym ,t n,k the la w of the resulting random p erm utation. Recall that ν t n,k is a measure o v er A n or A c n , but we can regard it as a measure in S n where the extended part has zero measure. In this wa y , we write ∼ l \ ν t n,k as the pushforw ard measure of ν t n,k o v er ∼ l \ S n . Similarly , we write ∼ l \ ν sym ,t n,k for the pushforw ard measure of ν sym ,t n,k o v er ∼ l \ S n . W hen l ≥ 2 , it is clear that ∼ l \ ν t n,k and ∼ l \ ν sym ,t n,k are identical. Lemma 3.5. L et n = ml , such that 2 ≤ l ≤ n o (1) . Mor e over, supp ose that x = n o (1) . Sample a uniformly r andom subset T of size x in [ n ] , then sample a uniformly r andom element σ ∈ S [ n ] \ T and view it as an element of S n by fixing al l of the elements in T . Then, we have P ( σ is cluster e d ) ≤ n − 1+ o (1) , (3.1) and d T V ( | F n,k ( σ ) | , x + Pois( l )) ≤ n − 1+ o (1) . (3.2) Pr o of. First, we ha v e P ( | T ∩ T i,l | ≤ 1 ∀ 1 ≤ i ≤ m ) =  1 − l n   1 − 2 l n  · · ·  1 − ( x − 1) l n  = 1 − n − 1+ o (1) . (3.3) Second, for all 1 ≤ i ≤ m suc h that T ∩ T i,l = ∅ , w e hav e P ( F n,l ( σ ) ∩ T i,l  = ∅ ) =  1 − l n − x  · · ·  1 − l n − x − l + 1  = 1 − l 2 n − n − 2+ o (1) , so that P ( F n,l ( σ ) ∩ T i,l = ∅ ) = l 2 n + O ( n − 2+ o (1) ) . Also, we hav e E [ F n,l ( σ ) ∩ T i,l ] = l 2 n − x = l 2 n + O ( n − 2+ o (1) ) , 18 JIAHE SHEN and this implies P ( | F n,l ( σ ) ∩ T i,l | ≥ 2) ≤ n − 2+ o (1) . Summing up all 1 ≤ i ≤ m suc h that T ∩ T i,l = ∅ , w e hav e P ( | F n,l ( σ ) ∩ T i,l | ≥ 2 , ∀ i such that T ∩ T i,l = ∅ ) ≤ n − 1+ o (1) . (3.4) Third, for all 1 ≤ i ≤ m suc h that T ∩ T i,l  = ∅ , w e hav e P ( F n,l ( σ ) ∩ T i,l  = ∅ ) ≥  1 − l n − x  · · ·  1 − l n − x − l + 1  = 1 − l 2 n − n − 2+ o (1) = 1 − n − 1+ o (1) , and therefore P ( F n,l ( σ ) ∩ T i,l  = ∅ , ∀ i suc h that S t ∩ T i,l  = ∅ ) ≤ x · n − 1+ o (1) = n − 1+ o (1) . (3.5) The b ounds in (3.3), (3.4), and (3.5) together complete the pro of of (3.1). The pro of of (3.2) is a standard Poisson approximation argument; see, for example, the pap ers from Arratia-Goldstein- Gordon [1, 2]. □ No w, we are ready for our pro of of Theorem 3.1 and Theorem 3.2. Pr o of of The or em 3.1. In this case, we hav e t ′ = n ( c − 1 2 log l ) k = − n (log log n − ω (1)) 2 k b ecause l = o (log n ) . Therefore, applying the natural quotient map and using monotonicit y of total v ariation distance under pushforw ard, we can obtain from case (2) of Theorem 1.6 that d TV ( ∼ l \ P ∗ t n,k , ∼ l \ ν sym ,t n,k ) = o (1) . Hence it suffices to prov e d TV ( ∼ l \ ν sym ,t n,k , ∼ l \ U S n ) = 2Φ(exp( − c ) / 2) − 1 + o (1) . No w, let us sample ˜ σ ∈∼ l \ S n with resp ect to the law of ∼ l \ ν sym ,t n,k , and sample ˜ τ ∈∼ l \ S n with resp ect to the la w of ∼ l \ U S n . Observe that for all T ⊆ [ n ] , the conditional la w of ˜ σ given the even t F n,k ( ˜ σ ) = T is the same as the conditional law of ˜ τ under the ev en t F n,k ( ˜ τ ) = T . Therefore, it suffices to prov e d T V ( F n,k ( ˜ σ ) , F n,k ( ˜ τ )) = 2Φ(exp( − c ) / 2) − 1 + o (1) . Recall from our definition that γ t = exp( − c + 1 2 log l ) = √ l exp( − c ) . Hence, applying (3.2), we ha v e P ( | F n,k ( ˜ σ ) | ≥ log n ) ≤ n − 1+ o (1) + P  P ois  l + √ l exp( − c )  ≥ log n  = n − 1+ o (1) , P ( | F n,k ( ˜ τ ) | ≥ log n ) ≤ n − 1+ o (1) + P (Pois( l ) ≥ log n ) = n − 1+ o (1) . Therefore, by natural truncation it follows from (3.1) and (3.2) that d T V ( F n,k ( ˜ σ ) , F n,k ( ˜ τ )) = d T V ( | F n,k ( ˜ σ ) | , | F n,k ( ˜ τ ) | ) + o (1) = d T V  P ois  l + √ l exp( − c )  , Pois( l )  + o (1) = 2Φ(exp( − c ) / 2) − 1 + o (1) . (3.6) Here, the third line is a straightforw ard consequence of l = ω (1) , together with the one-crossing prop ert y and the classical lo cal limit approximation for Poisson la ws; see, for example, standard references on limit theorems such as Petro v [11, Chapter VI I]. □ Pr o of of The or em 3.2. Applying the natural quotient map and using monotonicity of total v ari- ation distance under pushforw ard, we obtain from case (1) of Theorem 1.6 that d TV ( ∼ l \ P ∗ t n,k , ∼ l \ ν sym ,t n,k ) = o (1) . Hence it suffices to prov e d TV ( ∼ l \ ν sym ,t n,k , ∼ l \ U S n ) = d TV (P ois( l + exp( − c )) , P ois( l )) + o (1) . THE k -CYCLE SHUFFLING WITH REPEA TED CARDS 19 No w, let us sample ˜ σ ∈∼ l \ S n with resp ect to the law of ∼ l \ ν sym ,t n,k , and sample ˜ τ ∈∼ l \ S n with resp ect to the law of ∼ l \ U S n . Observe that for all T ⊆ [ n ] , the conditional la w of ˜ σ under the ev en t F n,k ( ˜ σ ) = T is the same as the conditional law of ˜ τ given the ev en t F n,k ( ˜ τ ) = T . Therefore, it suffices to prov e d T V ( F n,k ( ˜ σ ) , F n,k ( ˜ τ )) = d TV (P ois( l + exp( − c )) , P ois( l )) + o (1) . Recall from our definition that γ t = exp( − c ) . Hence, applying (3.2), w e hav e P ( | F n,k ( ˜ σ ) | ≥ log n ) ≤ n − 1+ o (1) + P (Pois( l + exp( − c )) ≥ log n ) = n − 1+ o (1) , P ( | F n,k ( ˜ τ ) | ≥ log n ) ≤ n − 1+ o (1) + P (Pois( l ) ≥ log n ) = n − 1+ o (1) . Therefore, by natural truncation it follows from (3.1) and (3.2) that d T V ( F n,k ( ˜ σ ) , F n,k ( ˜ τ )) = d T V ( | F n,k ( ˜ σ ) | , | F n,k ( ˜ τ ) | ) + o (1) = d T V (P ois( l + exp( − c )) , P ois( l )) + o (1) . (3.7) This completes the pro of. □ References [1] Ric hard Arratia, Larry Goldstein, and Louis Gordon. T wo momen ts suffice for p oisson approximations: the Chen-Stein metho d. The Annals of Pr ob ability , pages 9–25, 1989. [2] Ric hard Arratia, Larry Goldstein, and Louis Gordon. P oisson approximation and the Chen-Stein method. Statistic al Scienc e , pages 403–424, 1990. [3] Nathanaël Berestyc ki, Oded Sc hramm, and Ofer Zeitouni. Mixing times for random k -cycles and coalescence- fragmen tation chains. 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