Eigenvalues of Random Wreath Products

Consider a uniformly chosen element $X_n$ of the $n$-fold wreath product $\Gamma_n = G \wr G \wr \cdots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $X_n$ in the natural $s^n$-dimensional permutation representation (the composition representation) are investigated by considering the random measure $\Xi_n$ on the unit circle that assigns mass $1$ to each eigenvalue.  It is shown that if $f$ is a trigonometric polynomial, then  $\lim_{n \rightarrow \infty} P\{\int f d\Xi_n \ne s^n \int f d\lambda\}=0$, where $\lambda$ is normalised Lebesgue measure on the unit circle. In particular, $s^{-n} \Xi_n$ converges weakly in probability to $\lambda$ as $n \rightarrow \infty$.  For a large class of test functions $f$ with non-terminating Fourier expansions, it is shown that there exists a constant $c$ and a non-zero random variable $W$ (both depending on $f$) such that $c^{-n} \int f d\Xi_n$ converges in distribution as $n \rightarrow \infty$ to $W$.  These results have applications to Sylow $p$-groups of symmetric groups and autmorphism groups of regular rooted trees.


Introduction
Let T denote the regular rooted b-ary tree of depth n.That is, T is a tree with 1 + b + b 2 + + b n vertices such that one vertex (the root) has degree b, the b n leaf vertices have degree 1, and all other vertices have degree b + 1.
Consider the group ? of automorphisms of T.An element 2 ? is a permutation of the vertices of T such that the images of any two adjacent vertices (that is, two vertices connected by an edge) are again adjacent.

h h h h h h h h h
The group ? is nothing other than the n-fold wreath product of the symmetric group on b letters, S b , with itself.We recall the general de nition of a wreath product as follows.Let G and H be two permutation groups acting on sets of size s and t, respectively, which we will identify with f0; 1; : : :; s?1g and f0; 1; : : :; t?1g.As a set, the wreath product GoH of G and H is the Cartesian product G t H; that is, an element of GoH is a pair (f; ), where f is function from f0; 1; : : :; t?1g into G and 2 H. Setting f := f ?1 for f 2 G t and 2 H, the group operation on GoH is given by (f; )(f 0 ; 0 ) := (ff 0 ; 0 ), where multiplication is coordinatewise in G t .It is not hard to see that for three permutation groups G; H; K the group (G o H) o K is isomorphic to the group G o (H o K), and so it makes sense to refer to these isomorphic groups as G o H o K.More generally, it makes sense to speak of the wreath product G 1 o G 2 o o G n of n permutation groups G 1 ; G 2 ; : : :; G n .
Excellent references for wreath products with extensive bibliographies are Ker71, Ker75, JK81].Besides their appearance as the automorphism groups of regular rooted trees, wreath products are important in the representation theory of the symmetric group and in various problems arising in the Polya{Red eld theory of enumeration under group action.Classically, they appeared in the work of Cauchy on Sylow p-groups of the symmetric group.For example, the Sylow p-group of S p r , the symmetric group on p r letters, is the r-fold wreath product C p o C p o o C p , where C p is the cyclic group of order p.The Sylow p-group of S n for a general n is a certain product of such groups (see 4.1.22 of JK81]).
For G and H as above, there is a natural representation of G o H as a group of permutations of the set f0; 1; : : :; t?1g f0; 1; : ::; s?1g.In this permutation representation, the group element (f; ) 2 GoH is associated with the the permutation that sends the pair (i 0 ; i 00 ) to the pair (j 0 ; j 00 ) where j 0 = (i 0 ) and j 00 = f( (i 0 ))(i 00 ).
Either of these representations is called the composition representation.
Let X n be a uniform random pick from the n-fold wreath product ?n := G o G o o G.The random group element X n will have a corresponding composition representation M n .If we wished to describe the distribution of the s n s n random matrix M n , we would need to specify the order in which the successive \wreathings" were performed.However, two di erent orders produce matrices that are similar (with the similarity e ected by a permutation matrix), and so the eigenvalues of the composition representation of X n (and their multiplicities) are well-de ned without the need for specifying such an order.Let n denote the random discrete measure of total mass s n on the unit circle T C that is supported on this set of eigenvalues and assigns a mass to each eigenvalue equal to its multiplicity.We will be interested in the asymptotic behaviour of the measure n .In particular, we will investigate the behaviour of the integrals R T f d n for suitable test functions f.
and so the behaviour of R T f d n for a function f with Fourier expansion f(z) = P 1 k=?1 c k z k is determined by the behaviour of the random variables T n;k := Tr(M k n ), k 1.Let S n;k denote the number of k-cycles in the composition representation of X n .By a standard fact about permutation characters (see, for example, 6.13 of Ker75]), and hence, by M obius inversion, where is the usual M obius function (?1) j ; if i is the product of j distinct primes, 0; otherwise.
Therefore, it is equally useful to study the random variables S n;k , k 1.
Example 1.3.Consider the n-fold wreath product S 2 o S 2 o o S 2 , that is, the group of automorphisms of the regular rooted binary tree of depth n (a group of order 2 n ).It follows from Lemma 2.3 below that the cycle count S n;k is 0 unless k is of the form 2 j , 0 j n.Observe from (1.2) that if k = 2 h r where 2 -r, then T n;k = X 2 j jk;j n 2 j S n;2 j = X 2 j j2 h ; j n 2 j S n;2 j = T n;2 h^n: It thus su ces to understand the random variables T n;2 h, 0 h n.A simulated realisation of the random group element X 6 resulted in the eigenvalues shown (with multiplicities) in Figure 1.3.

B B B B B B B B B B B B B B B B B B B B B @
Using the facts we develop below in Section 3, it is not di cult to show in this example that E T n;1 ] = 1 and E T n;2 ] = n + 1 (in general, E S n;p ] = n p and E T n;p ] = 1 + np p for a prime p).However, 5 realisations out of 11 resulted in the value 0 for both T 6;1 and T 6;2 .This suggests that for large n the random variables T n;1 and T n;2 take the value 0 with probability close to 1, while the expectation is maintained by large values being taken with probability close to 0. The following result (proved in Section 3) shows that this is indeed the case.
Notation 1.4.Let denote Lebesgue measure on the unit circle normalised to have total mass 1.
Theorem 1.5.For a trigonometric polynomial f(z In particular, the random probability measure s ?n n converges weakly in probability to as n ! 1. converges in distribution as n ! 1 to a random variable cW P 1 k=1 d k , where 0 < W < 1 almost surely.Remark 1.9.i) Condition (b) of Theorem 1.8 can be modi ed to the weaker condition d k 0 (with a corresponding modi cation in the conclusion).The modi cation is discussed after the proof of the theorem in Section 4. ii) Suppose that (d k ) 1 k=1 is an arbitrary positive multiplicative sequence.Note that d 1 = 1 (by the multiplicative assumption), 1 = 1 (by Burnside's Lemma and the assumption that G acts transitively { see Section 3), and k > 0 for some k 2 (again by transitivity).Thus P s k=1 kd k k > 1 and ( P s k=1 kd k k ) 2 > P s k=1 kd k k .For any group G the condition (d) of Theo- rem 1.8 is therefore implied by the condition kd k 1 for all k.
iv) If G is an arbitrary nite group of order s acting on itself via the regular representation, then k = !k =k, kjs, where !k is the number of elements in G with order k.
We end this introduction with some bibliographic comments on the substantial recent interest in eigenvalues of random matrices in general and eigenvalues of Haar distributed random matrices from various compact groups in particular.
A general reference to the history of random matrix theory and its applications is Meh91].Asymptotics for the traces of powers of unitary, orthogonal and symplectic matrices (equivalently, integrals of powers against the analogue of the measure n ) are investigated in DS94] (see also Rai97]).Integrals of more general well-behaved functions against the analogue of n for these groups are studied in Joh97].The number of eigenvalues in an interval for the unitary group (that is, the integral of an indicator function against the analogue of n ) is investigated in Wie98].The logarithm of the characteristic polynomial of a random unitary matrix is also the integral of a suitable function against the analogue of n , and this object is the subject of HKO00, KS00a, KS00b].A general theory for the unitary, orthogonal and symplectic groups that subsumes much of this work is presented in DE01].
Random permutations give rise to random permutation matrices.Given the connection between cycle counts of permutations and traces of the corresponding matrices, some of the huge literature on the cycle structure of uniform random permutations can be translated into statements about eigenvalues of random permutation matrices.More in the spirit of this paper, the number of eigenvalues in an interval and the logarithm of the characteristic polynomial are investigated in Wie00] and HKOS00], respectively.The former paper treats not only the symmetric group, but also the wreath product of a cyclic group with a symmetric group.
There is a limited literature on other probabilistic aspects of wreath products.As mentioned above, the Sylow p-group of S p r is a wreath product.The distribution of the order of a random element of this group is studied in PS83b], while the distribution of the degree of a randomly chosen irreducible character is studied in

Useful facts
The following is obvious and we leave the proof to the reader.?j (j ) : : : ` ?1 (j ) ; that is, can be decomposed into c( ) cycles, with the th cycle of length ` .The elements of G de ned by g (f; ) := f(j )f( ?1 (j )) : : :f( ?(` ?1) (j )) = ff : : :f `nu?1 (j ) are called the cycle products of (f; ).Note that the de nition of g (f; ) depends on the choice of the cycle representative j , so to give an unambigious de nition we would need to specify how j is chosen (for example, as the smallest element of the cycle).However, di erent choices of cycle representative lead to conjugate cycle products (see 4.2.5 of JK81]).

Proof of Theorem 1.5
In order to prove the theorem, it su ces by (1.1) to show that lim n PfT n;k 6 = 0g = 0 for all k 1: By (1.2), it su ces in turn to show that (3.1) lim n PfS n;k 6 = 0g = 0 for all k 1: We will now choose a speci c order of the successive \wreathings" in the construction of ?n = GoGo oG that leads to a useful inductive way of constructing X 1 ; X 2 ; : : : on the one probability space.Take ?n = G o (G o (G o ( o G) : : :)).In other words, think of ?n as a permutation group on a set of size s n and build ?n+1 as G o ?n .Start with X 1 as a uniform random pick from G. Suppose that X 1 ; X 2 ; : : :; X n have already been constructed.Take X n+1 to be the pair (F; X n ), where F is a G s n -valued random variable with coordinates that are independent uniform random picks from G which are also independent of X n .It follows inductively from Lemma 2.1 that X n+1 is a uniform random pick from ? n+1 .
It is immediate from Lemma 2.1 that the cycle products of (F; X n ) consist of products of disjoint collections of the independent uniformly distributed G-valued random variables F(j).The segregation of the F(j) into the various cycle products is dictated by the independent ?n -valued random variable X n .Therefore, conditional on X n , the cycle products of (F; X n ) form a sequence of independent, uniformly distributed G-valued random variables.
Put S 01 := 1 and S 0k := 0, k > 1.By Lemma 2.3, the stochastic process ((S n;k ) 1 k=1 ) 1 n=0 taking values in the collection of in nite-length integer-valued sequences is thus a Galton{Watson branching process with in nitely many types (the types labelled by f1; 2; 3; : : :g).An individual of type k can only give birth to individuals of types k; 2k; 3k; : ::.Moreover, the joint distribution of the sequence of integer-valued random variables recording the number of o spring of types k; 2k; 3k; : :: produced by an individual of type k does not depend on k and is the same as that of the sequence recording the number of cycles of lengths 1; 2; 3; : : : for a uniformly chosen element of G.
Recall our standing assumption that G acts transitively.It follows from this and Burnside's Lemma (see, for example, Lemma 4.1 of Ker75]) that the number of 1-cycles (that is, xed points) of a uniformly chosen element of G is a non-trivial random variable with expectation 1 = 1.By the observations above, the process (S n;1 ) 1 n=0 is a critical (single-type) Galton{Watson branching process and hence this process becomes extinct almost surely.That is, if we set 1 := inffn : S n;1 = 0g, then Pf 1 < 1g = 1 and 0 = S 1 ;1 = S 1 +1;1 = : : :.By the observations above and the strong Markov property, (S 1 +n;2 ) 1 n=0 is also a critical (single-type) Galton{Watson branching process (with the same o spring distribution as (S n;1 ) 1 n=0 ) and so this process also becomes extinct almost surely.
Remark 3.1.Much of the work on eigenvalues of Haar distributed random matrices described in the Introduction is based on moment calculations.As noted in the Introduction, E T n;1 ] = 1 for all n, and so a result such as Theorem 1.5 could not be proved using such methods.

Proof of Theorem 1.8
By Theorem 1.5, we may suppose that c k = d k for all k.From equations (1.1) and (1.2) we have, in the notation of Section 3, that Z T `d`Sn;`! : Setting := P s j=1 jd j j and (W n ) 1 n=0 := ( ?n P 1 k=1 kd k S n;k ) 1 n=0 , it thus suf- ces to establish that W n converges in distributions as n ! 1 to a random variable W with Pf0 < W < 1g = 1.Construct X 1 ; X 2 ; : : : in the manner described in Section 3, so that ((S n;k ) 1 k=1 ) 1 n=0 is an in nitely{many{types Galton{Watson branching process.Let F n := fX 1 ; X 2 ; : : :; X n g and observe that `d`Sn;`! : Thus, (W n ) 1 n=0 is a nonnegative martingale with respect to the ltration (F n ) 1 n=0 , and hence W n converges almost surely as n ! 1 to an almost surely nite nonnegative random variable W.
We will next show that E W] = 1 by showing that the martingale (W n ) 1 n=0 is bounded in L 2 (P) (and hence converges in L 2 (P) as well as almost surely).By orthogonality of martingale increments, Let j 0 ;j 00 denote the covariance between the numbers of j 0 -cycles and j 00 -cycles in uniform random pick from G. By the branching process property, S n;`X j 0 ;j 00 `:j 0 d `:j 0 `:j 00 d `:j 00 j 0 ;j 00 `2d 2 `Sn;`!0 @ X j 0 ;j 00 j 0 d j 0j 00 d j 00 j 0 ;j 00 1 A : Note that the sequence (`2d 2 `)1 `=1 is multiplicative.Thus, setting " := P s j=1 j 2 d 2 j j , the sequence (" ?nP 1 k=1 k 2 d 2 k S n;k ) 1 n=0 is a martingale by the same argument that established (W n ) 1 n=0 was a martingale.Consequently, E (W n+1 ?W n ) 2 ] = ?2(n+1)" n X j 0 ;j 00 j 0 d j 0 j 00 d j 00 j 0 ;j 00: By assumption, 2 > ", and hence sup n E W 2 n ] < 1, as required.
For a partition a = (1 a1 ; 2 a2 ; : : :; s as ) of s (that is, a has a 1 parts of size 1, a 2 parts of size 2, et cetera and, in particular, P i ia i = s) let p(a 1 ; a 2 ; : : :; a s ) denote the probability that a uniformly chosen element of G has a 1 1-cycles, a 2 2-cycles et cetera.Write g(u 1 ; u 2 ; : : :; u s ) := where h(u) := g(u; : : :; u) is the probability generating function of the total number of cycles in a random uniform pick from G. The equation (4.1) has two solutions in the interval 0; 1]: namely, 1 and the probability of eventual extinction for a (single-type) Galton{Watson branching process with the distribution of the total number of cycles as its o spring distribution.Because E W] = 1, cannot be 1.The other root of (4.1) is clearly 0, because the total number of cycles is always at least 1.This completes the proof of the theorem.
Remark 4.1.Theorem 1.8 was proved under the hypothesis (b) that d k > 0 for all k 2 M. If this is weakened to the hypothesis that d k 0 for all k 2 M, then a similar result holds.Hypothesis (e) needs to be modi ed to an assumption that lim k!1;dk>0 c k =d k = c exists and d k = 0 implies c k = 0 for all k su ciently large.
The conclusion then becomes that the stated limit holds with 0 W < 1 almost surely.The probability PfW = 0g is the probability of eventual extinction for a Galton-Watson branching process with o spring distribution the total number of cycles in a random uniform pick from G having lengths in the set fk : d k > 0g.

Figure 1 .
Figure1.Eigenvalues for a random automorphism of the rooted binary tree of depth 6.
1 C C C C C C C C C C C C C C C C C C C C C A: The corresponding realisations of the cycle counts are 0 B B B B B B B B B B B B B B B B B B B B B B B B @ S 6;1 S 6;2 S 6;4 S 6;8 S 6;16 S 6;32 S 6 Theorem 1.5 leaves open the possibility of interesting behaviour for R T f d n for certain functions f having non-terminating Fourier expansion f(z) = P 1 k=?1 c k z k with c 0 = 0.Because of (1.1) it su ces to consider functions of the form f(z) = P 1 k=1 c k z k .De nition 1.6.A complex sequence (d k ) 1 k=1 is multiplicative if d k:`= d k d `.Obvious examples of multiplicative sequence are d k = k for 2 C .In general, a multiplicative function is speci ed by assigning arbitrary values of d p to each prime p.The value of d k for an integer k with prime decomposition k = p h1 1 p h2 2 : : :p hm m is then d h1 p1 d h2 p2 : : :d hm pm .Notation 1.7.Let k denote the expected number of k-cycles in the cycle decomposition of a permutation chosen uniformly at random from G. Write M for the smallest subset of N that contains f1 k s : k > 0g and is closed under multiplication.The following result is proved in Section 4. Theorem 1.8.Consider two sequences (c k ) 1 k=1 and (d k ) 1 k=1 that satisfy the following conditions: a) (d k ) 1 k=1 is multiplicative, b) d k > 0 for all k such that k > 0, c) P 1 k=1 d k < 1, d) ( P s k=1 kd k k ) 2 > P s k=1 k 2 d 2 k k , e) lim k!1; k2M c k =d k = c exists.Then the sequence of random variables ( lim k!1 c k =k = c exists.Then the sequence of random variables ( k n (dz) converges in distribution as n ! 1 to a random variable cW P 1 k=1 k , where 0 < W < 1 almost surely.Example 1.11.For the reader's bene t, we record the expected cycle counts k in some examples (see 5.16 of Ker75]).i) If G = S s , the symmetric group of order s! acting on a set of size s, then k = k ?1 , 1 k s.ii)If G = C s , the cyclic group of order s acting on a set of size s) := #f1 ` k : (`; k) = 1g is the Euler function.
following result is 4.2.19 in JK81].