Absolute continuity of the limiting eigenvalue distribution of the random Toeplitz matrix

We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schr\"{o}dinger operators. The similar question for Hankel matrices remains open.


Introduction
An n × n symmetric random Toeplitz matrix is given by counting multiplicity. Bryc, Dembo and Jiang (2006) established using method of moments that with probability 1, µ(n −1/2 T n ) converges weakly as n → ∞ to a nonrandom symmetric probability measure γ which does not depend on the distribution of a 0 , and has unbounded support. They conjecture (see Remark 1.1 there) that γ has a smooth density. In this note, we give a partial solution: Theorem 1. The measure γ is absolutely continuous with density bounded by 8.
The actual bound we get is 16 √ 2 π = 7.20 . . ., but we do not expect it to be optimal.
It seems that the method of moments is of little use in determining the existence of the absolute continuity of the limiting eigenvalue distribution. Indeed our proof goes along a completely different path. We make use of the fact that the spectrum of the Gaussian Toeplitz matrix can be realized as that of some diagonal matrix consisting of independent Gaussians conjugated by an appropriate projection matrix -a fact observed in a recent paper Sen and Virág (2011). The next key ingredient of our proof is a spectral averaging technique (Wegner type estimate) developed by Combes, Hislop and Mourre (1996) in connection to the problem of localization for certain families of random Schrödinger operators.
Our proof does not establish further smoothness property of γ. The absolute continuity for the limiting distribution of random Hankel matrices also remains open.

Connection between Toeplitz and circulant matrices
Since γ does not depend on the distribution of a 0 , from now on, we will assume, without any loss, that (a i ) i≥0 are i.i.d. standard Gaussian random variables. The remainder of the section we recall some facts about the connection between Toeplitz matrices and circulant matrices from Sen and Virág (2011). Let T • n be the symmetric Toeplitz matrix which has √ 2a 0 on its diagonal instead of a 0 . It can be easily shown (e.g. using Hoffman-Wielandt inequality, see Bhatia (1997)) that this modification has no effect as far as the limiting eigenvalue distribution is concerned.
The circulant matrix can be easily diagonalized as where U 2n is the discrete Fourier transform, i.e. a unitary matrix given by Clearly, d j = d 2n−j for all n < j < 2n. Also, (d j ) 0≤j≤n are independent mean zero Gaussian random variables with Var(d j ) = 1 if 0 < j < n and Var(d j ) = 2 if j ∈ {0, n}. Define Check that P 2n is a Hermitian projection matrix with P 2n (j, j) = 1/2 for all j. For notational simplification, we will drop the subscript 2n from the relevant matrices unless we want to emphasize the dependence on n.

Proof of the main theorem
For a vector u ∈ C m , let σ(A, u) be the spectral measure of matrix A at u. For a finite measure ν on R, its Cauchy-Stieltjes transform is given by Lemma 2. Let (e j ) 0≤j≤2n−1 be the coordinate vectors of R 2n . Then Before we start proving the above lemma, we state a simple fact about spectral measures of Hermitian matrices without proof.
Next we will prove a key lemma about the uniform bound on the Stieltjes transform of the expected empirical eigenvalue distribution of Toeplitz matrices.
Lemma 4. For all n, we have The above lemma will be a direct consequence of the following result of Combes et al. (1996) on the spectral averaging for one parameter family self-adjoining operators.
Proposition 5 (Combes et al. (1996)). Let H λ , λ ∈ R be a C 2 -family of self-adjoint operators such that D(H λ ) = D 0 ⊂ H ∀λ ∈ R, and such that (H λ − z) −1 is twice strongly differentiable in λ for all z, Im(z) = 0. Assume that there exist a finite positive constant c 0 , and a positive bounded self-adjoint operator B such that, on D 0 Also assume H λ is linear in λ, i.e.,Ḧ λ := d 2 H λ dλ 2 = 0. Then for all E ∈ R and twice continuously differentiable function g such that g, g ′ , g ′′ ∈ L 1 (R) and for all ϕ ∈ H, Proposition 5 is an immediate corollary of Theorem 1.1 of Combes et al. (1996) where instead ofḦ λ = 0, it was assumed that |Ḧ λ | ≤ c 1Ḣλ . The vanishing second derivative assumption shortens the the proof by a considerable amount. We have included a proof of the above proposition in the appendix to make this paper self-contained and also to make constant in the bound (5) explicit.
Proof of Lemma 4. Set E j = e j e * j + e 2n−j e * 2n−j for 1 ≤ j < n, and E j = e j e * j for j ∈ {0, n}. Take B j = Pe j e * j P or Pe 2n−j e * 2n−j P for 1 ≤ j < n and B j = Pe j e * j P for j ∈ {0, n}.
Fix 0 ≤ j ≤ n. We apply Theorem 5 with H λ = P D + (λ − d j )E j P. In words, we replace d j and d 2n−j by λ in PDP to get H λ . Note that H λ is random self-adjoint operator which is a function of {d k : 0 ≤ k ≤ n, k = j}. Also, H λ is linear in λ and so,Ḧ λ = 0.
SinceḢ λ = PE j P ≥ B j = P(j, j) −1 B 2 j , the condition (4) is satisfied with B = B j and c 0 = 2 since P(j, j) = 1/2. Take g = φ j where φ j be the density of Z for 0 < j < n or the density of √ 2Z for j ∈ {0, n}, Z being a standard Gaussian random variable. It is easy to check that g 1 = 1, g ′ 1 ≤ 2 π , g ′′ 1 ≤ 2. Then plugging ϕ = e j or e 2n−j and B j = Pe j e * j P or Pe 2n−j e * 2n−j P in (5) and taking expectation w.r.t. the remaining randomness {d k : 0 ≤ k ≤ n, k = j}, we obtain sup z:Im(z)>0 The lemma is now immediate from (7) and Lemma 2.