CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK

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1 CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK JONATHAN NOVAK AND ARIEL SCHVARTZMAN Abstract. In this note, we give a complete proof of the cutoff phenomenon for the star transposition random walk on the symmetric group.. Introduction An ergodic random walk on a finite regular graph has uniform measure as its stationary distribution. In approaching this limit, certain models exhibit a cutoff : the total variation distance to uniform remains close to its initial value for a time, and then abruptly becomes tiny [6]. The goal of this note is to present a complete analysis of one such model the star transposition random walk on the symmetric group. Identify the symmetric group S(d) with its right Cayley graph, as generated by the two-cycles (.) ( d), (2 d),..., (d d). Imagine a particle positioned at a given vertex of S(d). Let the position of this particle evolve in discrete time, according to the following rule: at each tick of the clock, the particle either stays put or jumps to a neighbouring vertex, all events occurring with equal probability. The generators (.) are called star transpositions, see [8, 2, 3, 5], and we will refer to this stochastic process as the star transposition random walk. Our goal is to give a complete proof of the following statement. Theorem. The star transposition random walk on S(d) exhibits cutoff at time d log d. A precise formulation of this result is given in Theorem 2 below. Note that Theorem is well-known to probabilists, see e.g. [5, Chapter 3] and [6, 9.5]. However, we have been unable to locate a proof of its precise version, Theorem 2, in the literature. We felt that assembling a complete proof of the cutoff phenomenon for the star transposition random walk was worthwhile for several reasons. First, the details of the argument are non-trivial and deserve to be recorded. Second, the modern approach to representations of symmetric groups, as developed in [4] and expounded in [2], facilitates a streamlined argument which clarifies an important point: the non-centrality of this process is not a significant obstacle, and what really matters is that the walk is driven by an element of the Gelfand-Tsetlin algebra. Finally, the star transposition random walk is special: among all minimal sets of transpositions which generate the symmetric group, star transpositions induce the Cayley graph

2 2 JONATHAN NOVAK AND ARIEL SCHVARTZMAN with largest spectral gap, and hence facilitate the fastest mixing. This fact is perhaps best understood in the framework of a recently resolved conjecture of Aldous; see [, 4, 0]..0.. Acknowledgement. This paper was written while the second author was participating in MIT s Undergraduate Research Opportunities Program (UROP) under the guidance of the first author. We thank Guillaume Chapuy for several helpful suggestions concerning the upper bound argument. 2. Precise statement Equip CS(d), the complex group algebra of S(d), with the inner product in which the permutations form an orthonormal basis, so that A = g S(d) g, A g for any A CS(d), and let be the corresponding l -norm, A = g S(d) g, A. Consider the unit vectors P, U CS(d) defined by with I the identity permutation and P = d (I + J), U = d! g S(d) g, (2.) J = ( d) + + (d d). Then P r is the distribution of the star transposition walk at time r, i.e. the inner product g, P r is the probability that the particle is located at vertex g of the Cayley graph after r jumps. Similarly, U is the uniform distribution on S(d), and P r U is the total variation distance between the distributions P r and U. We will prove the following precise version of Theorem. Theorem 2. Fix a large positive integer d. For any t > 0 such that r = d log d+td is an integer, P r U e 2 t. For any 0 < t < log d such that r = d log d td is an integer, P r U 400e t. 3. Upper bound To obtain the upper bound in Theorem 2, we will modify the arguments presented in [5, Chapter 3] and [2, Chapter 0], which treat the all-transpositions random walk. In the interest of brevity, we assume familiarity with the representation theory of finite groups in general, and the symmetric group in particular. We will use the following notation: Y (d) is the set of Young diagrams with d cells, T (d) is the set of standard Young tableaux with d cells, sh : T (d) Y (d) is the forgetful

3 CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK 3 map which sends each tableau to its underlying shape. Let ω(t ) denote the column index less the row index of the cell containing d in the tableau T. Let (V λ, ρ λ ), λ Y (d), be a complete set of pairwise non-isomorphic irreducible representations of CS(d). Applying the Diaconis-Shahshahani upper bound lemma [5, Chapter 3], we have (3.) P r U 2 2 λ (d) (dim λ) Tr ρ λ (P 2r ), where dim λ denotes the vector space dimension of V λ. In [4], a basis (3.2) v T, T sh (λ) of V λ is constructed such that (3.3) ρ λ (J)v T = ω(t )v T. The basis (3.2) is called the Gelfand-Tsetlin basis in [4]. The commutative subalgebra of CS(d) consisting of elements which act diagonally on the Gelfand-Tsetlin basis in each irreducible representation is known as the Gelfand-Tsetlin subalgebra. The generator of the star transposition random walk, P = I + J, belongs to the Gelfand-Tsetlin subalgebra. Indeed, it follows from (3.3) that the operator ρ λ (P ) is diagonal in the basis {v T } of V λ, and has spectrum ( + ω(t )), d T sh (λ). The spectrum of P is also computed in [9, Theorem 3.7], where it is used to study the hitting time of the star transposition random walk. From the above discussion, we conclude that Tr ρ λ (P 2r ) = T sh (λ) and hence the upper bound in (3.) becomes ( + ω(t )) 2r, (3.4) P r U 2 2 = 2 λ (d) ω(t )<d dim λ T sh (λ) ( + ω(t )) 2r dim sh(t )( + ω(t )) 2r. Let us decompose the right hand side of (3.4) as

4 4 JONATHAN NOVAK AND ARIEL SCHVARTZMAN (3.5) = = ω(t )<d 0 ω(t )<d 0 ω(t )<d dim sh(t )( + ω(t )) 2r dim sh(t )( + ω(t )) 2r + dim sh(t )( + ω(t )) 2r + (d ) ω(t )<0 (d ) ω(t )<0 dim sh(t )( + ω(t )) 2r dim sh(t )( ω(t ) ) 2r. In words, the first group of terms above corresponds to standard Young tableaux in which the cell containing d lies on or above the main diagonal, while the second group of terms corresponds to tableaux in which the cell containing d lies below the main diagonal. Given T T (d), let T denote its transpose, which is again a standard Young tableau. Noting that dim sh(t ) = dim sh(t ) and ω(t ) = ω(t ), the summation over tableaux with negative ω(t ) may be written (3.6) (d ) ω(t )<0 (3.7) We now argue that 0 ω(t ) d dim sh(t )( ω(t ) ) 2r = = dim sh(t )( ω(t )) 2r 0<ω(T ) d 0<ω(T ) d 0 ω(t ) d 0 ω(t )<d dim sh(t )( ω(t )) 2r dim sh(t )( ω(t )) 2r dim sh(t )( ω(t )) 2r. dim sh(t )( + ω(t )) 2r. While it is clear that each term of the sum on the left side of (3.7) is bounded by the corresponding term of the sum on the right hand side, the left sum has one more term, and so the inequality is not obvious. Consider the terms corresponding to the tableaux (3.8) and d 2 (3.9) d. On one hand, the tableau (3.8) contributes

5 CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK 5 (3.0) dim(d, )( (d 2)) 2r = (d )(d 3) 2r to the left side of (3.7), while the tableau (3.9) contributes (3.) ( (d )) 2r = (d 2) 2r. On the other hand, (3.8) contributes (3.2) (d )( + (d 2)) 2r = (d ) 2r+ to the right hand side of (3.7). Hence (3.7) is equivalent to the inequality (3.3) (d )(d 3) 2r + (d 2) 2r (d ) 2r+, which is easy to verify. We conclude that the upper bound (3.4) implies the upper bound (3.4) P r U 2 0 ω(t )<d dim sh(t )( + ω(t )) 2r We now use (3.4) to prove the upper bound in Theorem 2. From (3.4), we obtain (3.5) P r U 2 = 0 ω(t )<d λ (d) λ (d) λ (d) dim λ dim λ dim sh(t )( + ω(t )) 2r T sh (λ) ω(t ) 0 T sh (λ) ω(t ) 0 λ 2r (dim λ) 2. ( + ω(t )) 2r ( + λ ) 2r Let us partition the terms of the final sum in (3.5) according to initial row length: (3.6) λ (d) d λ 2r (dim λ) 2 = λ =d j d = ( j d )2r ( j d )2r (dim λ) 2 λ =d j (dim λ) 2.

6 6 JONATHAN NOVAK AND ARIEL SCHVARTZMAN Given a Young diagram λ with d cells in total and d j cells in the first row, write λ = (d j, λ ), where λ is the Young diagram with j cells obtained by removing the first row of λ. We then have the inequality (3.7) dim(d j, λ ) ( ) d dim λ. d j Indeed, there are ( d d j ) standard fillings of the first row (the initial cell must be filled with ), and dim λ standard fillings of the remaining cells. We thus arrive at the bound (3.8) d P r U 2 d = ( j ) 2r ( ) 2 d (dim λ ) 2 d d j λ Y (j) ) 2 j!, ( j ) 2r ( d d d j where the equality is a consequence of Frobenius s theorem that the sum of the squares of the dimensions of the irreducible representations of a finite group is equal to the cardinality of the group. Applying the inequality x e x, the bound (3.8) yields (3.9) d ( ) 2 P r U 2 e 2jr d d j! d j d = e 2jr d ( (d )! ) 2 (d j)! j! Now let t be a positive number such that d log d + td is an integer. Then we have (3.20) d ( ) 2 (d )! P d log d+td U 2 e 2jt d 2j (d j)! j! d ( ) 2 e 2t (d )! j! d j (d j)! d ( = e 2t (d ) j j! d j d e 2t j! e 2t. ) 2 Taking square roots yields the upper bound of Theorem 2.

7 CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK 7 For any E S(d), we have 4. Lower bound (4.) P r U Π E (P r ) Π E (U) Π E(P r ) Π E (U), where Π E End CS(d) acts on permutations according to (4.2) Π E (g) = { g, if g E 0, if g E. Thus, in order to obtain a lower bound on P r U, it suffices to construct a set E such that Π E (U) and Π E (P r ) are far apart. The function from sets to numbers defined by (4.3) E Π E (P r ) is the distribution of the star transposition random walk at time r. For small r, this distribution is concentrated in a small neighbourhood of the starting point of the random walk. Thus, assuming without loss in generality that the point of departure is the identity permutation, when r is small this distribution is concentrated on permutations with a large number of fixed points. Thus, for small r, the probability Π E (P r ) is small if E is a set of permutations whose members have a small number of fixed points. But a uniformly random permutation has few fixed points with high probability, and thus the difference Π E (U) Π E (P r ) will large in absolute value when r is small. The above strategy can be implemented algebraically using the fact that the character of the standard representation (V (d,), ρ (d,) ) of CS(d) is a fixed point count: (4.4) Tr ρ (d,) (g) = #{fixed points of g}. Set (4.5) χ (d,) = g S(d) Tr ρ (d,) (g)g, so that χ (d,), P r = Tr ρ (d,) (P r ) is one less than the expected number of fixed points of the star transposition random walk at time r. The operator ρ (d,) (P r ) has the eigenvalue (4.6) ( d) r with multiplicity dim V (d 2,) = d 2, corresponding to Gelfand-Tsetlin basis vectors in V (d,) indexed by Young tableaux of the form (4.7)... d.

8 8 JONATHAN NOVAK AND ARIEL SCHVARTZMAN It also has a one-dimensional kernel corresponding to the Gelfand-Tsetlin basis vector indexed by the tableau (4.8) Thus, for any positive integer r, (d ). d (4.9) χ (d,), P r = Tr ρ (d,) (P r ) = (d 2)( d )r is one less than the expected number of fixed points of the star transposition random walk at time r. For 0 < s χ (d,), P r, consider the event (4.0) E s = {g S(d) : Tr ρ (d,) (g) s}. By definition of E s, we have (4.) E s {g S(d) : Tr ρ (d,) (g) χ (d,), P r χ (d,), P r s}, so that (4.2) Π Es (P r ) (χ(d,) ) 2, P r χ (d,), P r 2 ( χ (d,), P r s) 2, by Chebyshev s inequality. As with any representation, the tensor square of the standard representation decomposes into the direct sum of the symmetric square and the alternating square: (4.3) V (d,) V (d,) = Sym 2 V (d,) Alt 2 V (d,). The alternating square of the standard representation is irreducible and isomorphic V (d 2,,) (see [, 3.2]). The symmetric square is reducible and decomposes as (4.4) Sym 2 V (d,) = V (d) V (d,) V (d 2,2), see [, Exercise 4.9]. Consequently, (4.5) (χ (d,) ) 2 = χ (d) + χ (d 2,2) + χ (d 2,,) + χ (d,), and our Chebyshev bound (4.2) becomes (4.6) Π Es (P r ) + χ(d 2,2), P r + χ (d 2,,), P r + χ (d,), P r χ (d,), P r 2 ( χ (d,), P r s) 2 = + Tr ρ(d 2,2) (P r ) + Tr ρ (d 2,,) (P r ) + Tr ρ (d,) (P r ) ( Tr ρ (d,) (P r ) ) 2 (Tr ρ (d,) (P r ) s) 2. The operator ρ (d 2,2) (P r ) has the eigenvalue

9 CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK 9 ( 2 ) r with multiplicity dim V (d 3,2) (d )(d 4) = d 2 corresponding to Gelfand-Tsetlin basis vectors in V (d 2,2) labelled by standard tableaux of the form as well as the eigenvalue... d, d r with multiplicity dim V (d 2,) = d 2 corresponding to basis vectors labelled by tableaux of the form.... d The operator ρ (d 2,,) (P r ) has the eigenvalue ( 2 ) r with multiplicity dim V (d 3,,) = (d )(d 2) d 2 corresponding to Gelfand-Tsetlin basis vectors in V (d 2,,) labelled by standard Young tableaux of the form as well as the eigenvalue... d, ( ) r d r with multiplicity dim V (d 2,) = d 2 corresponding to basis vectors labelled by tableaux of the form... d. It follows from the above computations that (4.7) χ (d 2,2), P r + χ (d 2,,), P r χ (d,), P r 2, so that our Chebyshev bound (4.2) implies (4.8) Π Es (P r ) + Tr ρ(d,) (P r ) (Tr ρ (d,) (P r ) s) 2. It remains only to make this upper bound explicit. We have Writing Tr ρ (d,) (P r ) = (d 2)( d )r = (d 2)e r log( d ).

10 0 JONATHAN NOVAK AND ARIEL SCHVARTZMAN this becomes log( x) = x x2 2 q(x), q(x) = + 2 x n n + 2, n= Tr ρ (d,) (P r ) = (d 2)e r( d + 2d 2 q( d )), so that, for any real t such that d log d td is a positive integer, Tr ρ (d,) (P d log d td ) = (d 2)e (d log d td)( d + 2d 2 q( d )) log d t (log d t+ = (d 2)e 2d q( d )) = ( 2 d )et e t log d 2d q( d ). Now, q( d ) is a positive, decreasing function of d N 2 with (4.9) q( 2 ) + 2 n= 2 n = 3 and q( d ) as d. Thus, for 0 t < log d, we have (4.20) e log d d q( d ) e t log d 2d q( d ) <, and this yields the two-sided inequality (4.2) 0.e t Tr ρ (d,) (P d log d td ) e t for d 3. From (4.8), we thus have (4.22) Π Es (P d log d td ) + e t (0.e t s) 2 for any 0 t < log d such that d log d td is a positive integer. Choosing s = 0.05e t, we obtain (4.23) Π E0.05e t (P d log d td ) + et e 2t = 400e t + 400e 2t. Let us now compare the above to the probability (4.24) Π E0.05e t (U). First, one less than the expected number of fixed points in a uniformly random permutation is (4.25) χ (d,), U = d! χ(d,), χ (d) = 0, by the orthogonality of irreducible characters. Using the tensor square decomposition described above together with orthogonality, it is easy to compute a second moment:

11 CUTOFF FOR THE STAR TRANSPOSITION RANDOM WALK (χ (d,) ) 2, U = d! χ(d), χ (d) + d! χ(d,), χ (d) + d! χ(d 2,2), χ (d) + d! χ(d 2,,), χ (d) We thus have (4.26) =. Π E0.05e t (U) = Π S(d)\E0.05e t (U) = Π {g S(d):(Tr ρ (d,) (g)) 2 >0.0025e 2t }(U), by Markov s inequality e2t = 400e 2t. Combining our estimates on Π E (P d log d td ) and Π E 2 3 et (U), we conclude 2 et 3 that (4.27) Π E0.05e t (U) Π E0.05e t (P d log d td ) 400e t, which is precisely the lower bound in Theorem 2. References. P. Caputo, T. M. Liggett, T. Richthammer, Proof of Aldous spectral gap conjecture, J. Amer. Math. Soc. 23 (200), T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Harmonic Analysis on Finite Groups, Cambridge Studies in Advanced Mathematics 08, T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras, Cambridge Studies in Advanced Mathematics 2, F. Cesi, On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions, J. Algebr. Comb. 32 (200), P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics Monograph Series, Volume, P. Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Natl. Acad. Sci. USA 93 (996), P. Diaconis, C. Greene, Applications of Murphy s elements, Stanford University Technical Report 335 (989). 8. V. Féray, Partial Jucys-Murphy Elements and Star Factorizations, European J. Combin. 33 (202), L. Flatto, A. M. Odlyzko, D. B. Wales, Random shuffles and group representations, Ann. Prob. 3 (985), J. Friedman, On Cayley graphs of the symmetric group generated by transpositions, Combinatorica 20 (2000), W. Fulton, J. Harris, Representation Theory: A First Course, Springer Graduate Texts in Mathematics 29, I. P. Goulden, D. M. Jackson, Transitive powers of Young-Jucys-Murphy elements are central, J. Algebra 32 (2009), J. Irving, A. Rattan, Minimal factorizations of permutations into star transpositions, Discrete Math. 309 (2009), A. Okounkov, A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 4 (996), I. Pak, Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers, and k-ary trees, Discrete Math. 204 (999), L. Saloff-Coste, Random walks on finite groups. 7. R. P. Stanley, Enumerative Combinatorics: Volume 2.

12 2 JONATHAN NOVAK AND ARIEL SCHVARTZMAN Department of Mathematics, Massachusetts Institute of Technology, USA address: Department of Mathematics, Massachusetts Institute of Technology, USA address:

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