Diagonal approximation of the form factor of the unitary group

G Berkolaiko

Department of Mathematics, Texas A&M University, College Station, TX 77840-3368, USA

Email: Gregory.Berkolaiko@math.tamu.edu

Received 21 September 2005, in final form 15 November 2005
Published 11 January 2006

Abstract. The form factor of the unitary group U(N) endowed with the Haar measure characterizes the correlations within the spectrum of a typical unitary matrix. It can be decomposed into a sum over pairs of `periodic orbits', where by periodic orbit we understand any sequence of matrix indices. From here the diagonal approximation can be defined in the usual fashion as a sum only over pairs of identical orbits. We prove that as we take the dimension N to infinity, the diagonal approximation becomes `exact', that is converges to the full form factor in the interval τ   [0, 1].

PACS numbers: 05.45.Mt, 02.10.De, 02.50.Ga, 05.10.Gg, 02.70.Hm

1. Introduction

One of the central questions in the quantum theory of classically chaotic systems is whether the fluctuations in the spectra follow the predictions of random matrix theory (RMT). The spectral form factor, which is the Fourier transform of the two-point correlation function, is often used to characterize these fluctuations.

The advantage of the form factor is that it has a convenient expansion in terms of pairs of periodic orbits of the classical system. This expansion is an application of trace formulae (see [1] for a review) and has been a starting point for many investigations. The progress of understanding the role of periodic orbits in the universal behaviour of the form factor of different systems is marked by such milestones as the diagonal approximation [2], the first off-diagonal contribution [3, 4] and, most recently, an outline of the complete expansion [5]. Quantum graphs were also used [6] as a mathematical toy model with a simple-to-understand periodic orbit structure and an exact trace formula.

In a recent paper, Degli Esposti and Knauf [7] proposed using the ultimate mathematical toy model for the random matrix conjecture - the random matrix theory itself. The authors argued that if one can perform a step-by-step expansion of the RMT form factor in terms of families of periodic orbits, it would greatly facilitate analogous expansions for other systems.

Degli Esposti and Knauf [7] studied the diagonal approximation to the form factor of one of the random matrix ensembles, the circular unitary ensemble (CUE). Their results, however, depend on a combinatorial conjecture, which is still unproven. The purpose of this letter is to present an alternative proof for their main result; our proof is unconditional and is significantly simpler. Our proof is based on the observation that the diagonal approximation can be expressed as a trace of certain doubly stochastic matrix. The asymptotic properties of the diagonal approximation can then be related to the spectral gap of the doubly stochastic matrix, which was previously studied by the author in [8].

The structure of the letter is as follows. In section 2, we introduce the circular unitary ensemble and the form factor. We follow by reviewing the main result of [7] and stating our main result in section 3. The relevant properties of doubly stochastic matrices are reviewed in section 4. Here, we also prove our main result which follows quite simply from a technical lemma on the asymptotics of the second largest eigenvalue. This lemma is proved in section 5 and we conclude by comparing our results with those of [7] in section 6.

2. The circular unitary ensemble and its form factor

The circular unitary ensemble is the unitary group U(N) equipped with the Haar measure:

Definition 1. CUE(N) is defined as the ensemble of all unitary N × N matrices endowed with the probability measure that is invariant under every automorphism

where V and W are any two N × N unitary matrices.

The form factor is a function describing the statistical properties of the spectrum of U. It is defined as the Fourier transform of the two-point spectral correlation function, R₂(x),

where exp(2πi_j) are the eigenvalues of U and the averaging is performed over the CUE(N). The form factor of the unitary ensemble is thus given by

The well-known formula for K_N(t) (see, e.g., [9, 10]) is

This formula can, in particular, be derived by expanding the trace in (3),

where the convention i_t + 1  =  i₁ is assumed here and throughout the letter. The above expression is analogous to the trace-formula expansions of the form factor in other systems, with i  =  (i₁,  ..., i_t) and j  =  (j₁,  ..., j_t) playing the role of the periodic orbits. It can be shown (see equation (8)) that only the terms with j being a permutation of i contribute to the form factor, i.e.

where S_t is the group of all t-permutations (the symmetric group). This step is analogous to only considering pairs of orbits with a small action difference. The approximation to (6) that restricts the double sum even further, to pairs of periodic orbits that are equal (up to a cyclic shift), is called the diagonal approximation

where the factor t comes from the number of possible shifts of an orbit. It is natural to expect the diagonal approximation to give the dominant contribution to K_N since it selects only those pairs of orbits i and j that produce a positive product in (6) for all choices of U. In this letter, we compare the form factor K_N(t) with its diagonal approximation.

The present letter was inspired by a study by Degli Esposti and Knauf [7]. They proved that, assuming validity of a certain combinatorial conjecture, the diagonal approximation K_1,N converges to t/N uniformly in t/N as N → ∞. The convergence is proved in a closed interval which is bounded away from zero. The combinatorial conjecture requires a more thorough introduction and will be delayed to a later section.

Here, we propose an alternative approach which exploits the fact that the sum in K_1,N is the trace of a doubly stochastic matrix. Our proof is unconditional, i.e. it does not rely on the conjecture, and the interval of convergence is extended.

3. Averages over the unitary group and the main result

The averages of products of the elements of matrices U have been studied by various authors [11-14]. Their main result is

where S_t is the symmetric group of permutations of the set {1,  ..., t}, δⁿ_k is the Kronecker delta and the coefficient V_N(σ ^- 1π) depends only on the lengths of cycles in the cycle expansion of σ ^- 1π.

To introduce the results of [7], we need the following definitions.

Definition 2. Let σ   S_t and i   {1, 2,  ..., t}. Then a cycle of i with respect to σ is the (finite) set . The cycle length of i is the number of elements in the cycle of i. The rank |σ| of a permutation σ is the number of disjoint cycles induced by σ. By |σ ∨ ρ| we denote the number of disjoint sets of the form . By τ we will denote the circular permutation (1, 2,  ..., t). If σ   S_t, σ ₊  will denote τ ^- 1στ.

Conjecture 1 (Degli Esposti and Knauf [7]). There exists a constant C₁ such that for all

Theorem 1 (Degli Esposti and Knauf [7]). Assuming conjecture 1 is true, the form factor K_N is approximated by the diagonal contribution in the following sense:

For all > 0

uniformly in τ  =  t/N   [, (1  -  )e/C₁].

Our main result is a version of the above theorem that drops the dependence on the conjecture and extends the interval of convergence.

Theorem 2. For all > 0

uniformly in τ  =  t/N   [,  ^- 1]. Thus, uniformly in τ  =  t/N   [, 1],

4. Doubly stochastic matrices and the proof of theorem 2

Definition 3. An entrywise non-negative N × N matrix M is called doubly stochastic if

The set of all such N × N matrices we denote by DS(N). DS(N) is a monoid with respect to the matrix multiplication, in particular if A and B are doubly stochastic then so is AB.

It follows directly from the definition that the vector e  =  (1/N,  ..., 1/N) is both a left and a right eigenvector of any matrix from DS(N). The corresponding eigenvalue 1 is the largest (by modulus) one. The following observation shows the importance of the asymptotics of the second largest eigenvalue.

If U is unitary then the matrix M defined by

is doubly stochastic. In fact, M is a so-called unistochastic matrix [15].

Definition 4. An N × N matrix M is called unistochastic if there exists a unitary matrix U satisfying (14). The set of unistochastic matrices is a proper subset of DS(N) when N > 2.

If M is given by (14) then

and, therefore, the diagonal approximation K_1,N is equal to

where the averaging is performed with respect to the measure induced on unistochastic matrices by the correspondence (14). The averages with respect to this measure were discussed in [8, 16]. In [8], it was speculated that the second largest eigenvalue of such random matrices M decays, on average, as N ^- 1/2. If this is indeed so, we immediately get

for t > 2, which would prove theorem 2. On the other hand, [16] conjectured that the difference Tr M^t_N  -  1 behaves like N^1 - t, which is also consistent with theorem 2. While the conjectures of [8, 16] are still unproven, we can use the technique of [8] to achieve the same result without directly calculating the eigenvalues of M.

Since the eigenvalues of M lie in the unit circle, it is intuitively clear that Tr M^t  =  ∑λ^t_i should be a decreasing function in t, at least `on average'. It is therefore tempting to show that Tr M^t_N → 1 for some fixed t as N → ∞ and then conclude that this property still holds if we send t to infinity as well. Indeed, this would work if the eigenvalues of M were positive real numbers, which is generically not the case for matrices M. To circumvent this difficulty, we form a matrix

which is symmetric, doubly stochastic and positive semidefinite (A ≥ 0). Now we can estimate the eigenvalues of M through the eigenvalues of A.

Proposition 1 ([8]). Second largest eigenvalues of M and A satisfy

Proof. The square roots of the eigenvalues of A defined by (18) are usually called the singular values of the matrix M. One of their fundamental properties is Weyl's majorization inequality [17], which implies

Since both M and A are stochastic matrices, λ₁(M)  =  λ₁(A)  =  1. Inequality (19) follows immediately.        

Now the idea is to find the power k such that λ(A)^k_N decay sufficiently fast. It turns out that k  =  3 is enough.

Lemma 1. The Haar probability measure on the unitary group induces a measure on symmetric doubly stochastic matrices A via (14) and (18). With respect to this measure λ₂(A)³_N  =  O(N ^- 2).

The proof is rather technical and will be given in a later section. This lemma enables us to give a simple proof of our main result.

Proof of theorem 2. We show that Tr M^t → 1. If τ ≥ and N is sufficiently large then t  =  τN ≥ 6, leading to

Now we use equation (16) to obtain the sought after conclusion.        

5. Proof of lemma 1

We start with the trace of A³,

since all eigenvalues of A are non-negative real numbers. Thus, we can estimate 0 ≤ λ₂(A)³ ≤ Tr A³  -  1. Writing out the trace we get

Now we average over the unitary group,

At this point, we can apply equation (8) with α_k  =  a_k and β_k  =  b_k with 1 ≤ k ≤ 6. The indices are a  =  (j, j, l, l, n, n) and b  =  (i, k, k, m, m, i). We denote

To cut down the number of the coefficients W that need to be calculated, we list some of the properties arising from the symmetries of (8).

Table 1. Factors contributing to Tr A3. We use notation D  =  (N  +  5)(N  +  4)(N  +  3)(N  +  2)(N  +  1)N. The remaining terms can be obtained using Wj,l,ni,k,m  =  Wi,k,mn,j,l. Whenever the first column lists three terms, the last column gives the number of occurrences of each of those terms.

Contributing terms	Notation	Value	Number of occurrences
W1,2,31,2,3	w33		N2(N  -  1)2(N  -  2)2
W1,2,21,2,3, W2,1,21,2,3, W2,2,11,2,3	w23		N2(N  -  1)2(N  -  2)
W1,2,21,2,2, W2,1,22,1,2, W2,2,12,2,1	w22a		N2(N  -  1)2
W1,2,22,1,2, W2,1,22,2,1, W2,2,11,2,2	w22b		N2(N  -  1)2
W1,2,22,2,1, W2,1,21,2,2, W2,2,12,1,2	w22c		N2(N  -  1)2
W1,1,11,2,3	w13		N2(N  -  1)(N  -  2)
W1,1,11,2,2, W1,1,12,1,2, W1,1,12,2,1	w12		N2(N  -  1)
W1,1,11,1,1	w11		N2

Thus, we get

where the factors of 2 arise due to W^j,l,n_i,k,m  =  W^i,k,m_n,j,l. The result is

which, together with inequality λ₂(A)³ ≤ Tr A³  -  1, finishes the proof.

6. Discussion

We presented an alternative and complete proof of the main result of [7]. However, the main importance of the results of [7] lies in the method of their derivation and in establishing a foothold in understanding the combinatorics of the form factor expansion. Below we formulate two conjectures which can be studied using the methods of [7].

The first conjecture is rephrasing an observation made in [8] regarding the asymptotics of the eigenvalues of unistochastic matrices M.

Conjecture 2. For any j ≥ 2 and k |λ_j(M)|^k  =  O(N ^- k/2), where the averaging is performed with respect to the measure induced by (14).

The second conjecture is a strengthening of our result which would follow if conjecture 2 were proved. It concerns the speed of convergence of the diagonal approximation and was first mentioned in [16].

Conjecture 3. For any τ   (0, ∞)

faster than any power of N.

Regarding possible extensions of our results, we expect our method to be directly applicable to other random matrix ensembles, such as circular orthogonal ensemble (COE). This is an ensemble of random unitary symmetric matrices S, with the measure induced by the Haar measure on U(N) via the factorization

We expect the matrix M (now symmetric) obtained from such S to also satisfy (30). The only significant difference from CUE is that the diagonal approximation in COE no longer recovers the full form factor (up to t  =  N) but just a leading (for small t/N) contribution to it. The proof can either proceed via factorization (31) or, more simply, by using the averages with respect to COE computed in [14].

Acknowledgments

The author is grateful to B Winn and a referee for many suggested improvements to the letter.

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