Circular law for random block band matrices with genuinely sublinear bandwidth

Random band matrices play an important role in mathematics and physics. Unlike many classical matrix ensembles, band matrices with small bandwidth are not of mean-field type and involve short-range interactions. As such, band matrices interpolate between classical mean-field models with delocalized eigenvectors (when the bandwidth is large) and models with localized eigenvectors and Poisson eigenvalue statistics (when the bandwidth is small).²² In addition, random band matrices have been studied in the context of nuclear physics, quantum chaos, theoretical ecology, systems of interacting particles, and neuroscience.^{3–6,29,49,50,83} Many mathematical results have been established for the eigenvalues and eigenvectors of random band matrices, especially Hermitian models; we refer the reader to Refs. 3, 8, 12, 13, 15, 16, 24, 27–29, 36, 38–41, 52–56, 58, 63, 64, 68, 74–76, 78, and 86 and references therein.

In this paper, we focus on non-Hermitian random block band matrices. Before we introduce the model, we define some notation and recall some previous results for non-Hermitian random matrices with independent entries. For an n × n matrix A, we let $λ1(A),…,λn(A)∈C$ denote the eigenvalues of A (counted with algebraic multiplicity). μ_A is the empirical spectral measure of A, defined as

where δ_z denotes a point mass at z.

The circular law describes the limiting empirical spectral measure for a class of random matrices with independent and identically distributed (iid) entries.

The circular law asserts that if X is an n × n iid random matrix with atom variable ξ having mean zero and unit variance, then the empirical spectral measure of $X/n$ converges almost surely to the uniform probability measure on the unit disk centered at the origin in the complex plane. This was proved by Tao and Vu in Refs. 80 and 81 and is the culmination of a large number of results by many authors.^{10,37,42,43,45,47,61,62} We refer the reader to the survey²⁰ for more complete bibliographic and historical details. Local versions of the circular law have also been established.^{7,25,26,85,87} The eigenvalues of other models of non-Hermitian random matrices have been studied in recent years; see, for instance, Refs. 1, 2, 14, 17–19, 31, 33–35, 44, 46, 57, 65–67, 69, 70, 72, and 84 and references therein.

Another model of non-Hermitian random matrices takes the form X ⊙ A, where the entries of the n × n matrix X are iid random variables with mean zero and unit variance and A is a deterministic matrix. Here, A ⊙ B denotes the Hadamard product of the matrices A and B, with elements given by (A ⊙ B)_ij = A_ijB_ij. The matrix A provides the variance profile for the model, and this model includes band matrices when A has a band structure. The empirical eigenvalue distribution of such matrices was studied in Ref. 34. For example, the following result from Ref. 34 describes sufficient conditions for the limiting empirical spectral distribution to be given by the circular law.

More generally, the results in Ref. 34 also apply to cases when conditions (1) and (2) are relaxed and the limiting empirical spectral measure is not given by the circular law. However, the results in Ref. 34, unlike the results in this paper, require the number of non-zero entries to be proportional to n² for the limit to be non-trivial.

In this paper, we focus on a model where the number of non-zero entries is polynomially smaller than n². We now introduce the model of random block band matrices we will study.

Note that each row and column of $X̃$ has 3b_n many nonzero random variables. Using the notation $[m]≔1,…,m$ for the discrete interval, we define

One motivation for the periodic block band matrix introduced above comes from theoretical ecology. Population densities and food webs, for example, can be modeled by a system involving a large random matrix.^6,59 The eigenvalues of this random matrix play an important role in the analysis of the stability of the system, and the circular law and elliptic law have previously been exploited for this purpose.⁶ It has been observed that many of these systems correspond to sparse random matrices with block structures (known as “modules” or “compartments”).^6,79 The periodic block band matrix introduced above is one such model with a very specific network structure.

Our main result below establishes the circular law for the periodic block band model defined above when b_n is genuinely sublinear. To the best of our knowledge, this is the first result to establish the circular law as the limiting spectral distribution for matrices with genuinely sublinear bandwidth.

We prove Theorem 1.4 by showing that there exist constants c, τ > 0 so that the empirical spectral measure of X converges to the circular law under the assumption that the bandwidth b_n satisfies cn ≥ b_n ≥ n^1−τ log n. In fact, the proof reveals that τ can be taken to be τ ≔ 1/33, as stated in Theorem 1.4, although this particular value can likely be improved by optimizing some of the exponents in the proof.

A few remarks concerning the assumptions of Theorem 1.4 are in order. First, the restriction on the bandwidth b_n ≥ n^1−τ log(n) with τ = 1/33 is of a technical nature and we believe that this condition can be significantly relaxed. For instance, we give an exponential lower bound on the least singular value of X − zI for $z∈C$ in Theorem 2.1 below. If this bound could be improved to say polynomial in n, then we could improve the value of τ to 1/2. It is possible that other methods could also improve this restriction even further. Second, the assumption that the entries have finite 4 + ϵ moments is due to the sublinear bandwidth growth rate. Our calculation requires higher moment assumptions for slower bandwidth growth, as can be seen from the Proof of Theorem 3.1.

A numerical simulation of Theorem 1.4 is presented in Fig. 1.

We use asymptotic notation under the assumption that n → ∞. The notations X = O(Y) and Y = Ω(X) denote the estimate |X| ≤ CY for some constant C > 0 and all n ≥ C. We write X = o(Y) if |X| ≤ c_nY for some c_n that goes to zero as n tends to infinity.

For convenience, we do not always indicate the size of a matrix in our notation. For example, to denote an n × n matrix A, we simply write A instead of A_n when the size is clear. We use b_n to denote the size of each block matrix and c_n ≔ 3b_n for the number of non-zero entries per row and column. We let $[n]≔1,2,3,…,n$ and e₁, e₂, …, e_n be the standard basis elements of $Cn$⁠. For a matrix A, a_ij will be the (i, j)-th entry, a_k will be the kth column, A^(k) represents the matrix A with its kth column set to zero, and $Hk$ will be the span of the columns of A^(k). Furthermore, A^* is the complex conjugate transpose of the matrix A, and when A is a square matrix, we let

where I denotes the identity matrix and $z∈C$⁠.

For the spectral information of an n × n matrix A, we designate

to be the eigenvalues of A (counted with algebraic multiplicity) and

to be the empirical measure of the eigenvalues. Here, δ_z represents a point mass at $z∈C$⁠. Similarly, we denote the singular values of A by

and the empirical measure of the squared-singular values as

Additionally, we use ‖A‖ to mean the standard ℓ₂ → ℓ₂ operator norm of A.

For a vector $v∈Cn$⁠,

Finally, we use the following standard notation from analysis and linear algebra. The set of unit vectors in $Cn$ will be denoted by Sⁿ⁻¹, i.e., $Sn−1≔v∈Cn:‖v‖=1$⁠, and the disk of radius r by $Dr≔z∈C:|z|<r$⁠. For any set $S⊂Cn$ and $u∈Cn$⁠,

|S| denotes the cardinality of the finite set S.

The rest of this paper is devoted to the Proof of Theorem 1.4. The proof proceeds via Girko’s Hermitization procedure (see Ref. 20), which is now a standard technique in the study of non-Hermitian random matrices. Following Ref. 54, we study the empirical eigenvalue distribution of $XzXz*$ for $z∈C$⁠. In particular, we establish a rate of convergence for the Stieltjes transform $XzXz*$ to the Stieltjes transform of the limiting measure in Sec. III. The key technical tool in our proof is a lower bound on the least singular value of X_z presented in Sec. II. In Sec. IV, following the method of Bai,¹⁰ these two key ingredients are combined and the Proof of Theorem 1.4 is given. The  Appendix contains a number of auxiliary results.

In this section, we present our key least singular value bound, Theorem 2.1. The crucial feature of our result is that the lower bound on the least singular value is only singly exponentially small in m. While this is most likely suboptimal and, indeed, we conjecture that our bound can be substantially improved, it is still significantly better than previous results in the literature. Notably, the work of Cook³² provides lower bounds on the least singular value for more general structured sparse random matrices; however, specialized to our setting, the lower bound there is doubly exponentially small in m [see Eq. (3.8) in Ref. 32], which only translates to a circular law for bandwidth (at best) Ω(n/log n).

We consider the translated periodic block band model X_z = X − zI, where X is as defined in (4) and $z∈C$ is fixed. Recall that m = n/b_n. Throughout this section, we will assume that b_n ≥ m ≥ m₀, where m₀ is a sufficiently large constant. Recall that for an n × n matrix A, we let s₁(A) ≥ s₂(A) ≥⋯ ≥ s_n(A) ≥ 0 denote its singular values.

Let us define the event

We begin by showing (Lemma 2.4) that $P(EKc)=O(1/cn)$⁠. This will allow us to restrict ourselves to the event $EK$ for the remainder of this section.

In order to bound the probability of the event $EKc$⁠, we will need the following two results on the smallest and largest singular values of (shifts of) complex random matrices with iid entries.

The next proposition can be readily deduced from Theorem 5.9 in Ref. 9 along with the standard Chernoff bound.

Applying the above two propositions [along with the triangle inequality for $‖(Di)z‖$] and using the union bound, we immediately obtain the following:

For the remainder of this section, we will restrict ourselves to the event $EK$⁠. For any $v∈Cn$⁠, we let

be the division of the coordinates into m vectors $v[i]∈Cbn$⁠. We will use v_i to denote the ith coordinate of v. For convenience, we use the convention that the indices wrap around, meaning, for example, that v_[m+1] = v_[1].

For α, β ∈ (0, 1), let

i.e., L_α,β consists of those unit vectors that have sufficiently many large coordinates. For us, α and β are constants depending on K, which will be specified later. Then, as $sn(Xz)=infv∈Sn−1‖Xzv‖$⁠, we can decompose the least singular value problem into two terms,

We begin with a lemma due to Rudelson and Vershynin, which converts the first term in (5) into a question about the distance of a random vector to a random subspace.

The distance of x_k − ze_k to $Hk$ can be bounded from below by $|⟨xk−zek,n̂⟩|$⁠, where $n̂$ is a unit vector orthogonal to $Hk$⁠. Our next goal is to obtain some structural information about any vector normal to $Hk$⁠. For convenience of notation, we will henceforth assume that k = 1; the same arguments are readily seen to hold for other values of k as well. Moreover, since the distribution of X_z is invariant under transposition, we may also assume that x₁ − ze₁ is the first row of X_z and that $H1$ is the subspace spanned by all the rows except for the first.

Recall that $H1$ is the subspace generated by all the rows of X_z except for the first row. The next proposition establishes that if v is normal to $H1$⁠, then there are sufficiently many v_[i] with large enough norm. Our approach to lower bounding the coordinates of v is similar to the methods used in Ref. 23; our proof is also similar in spirit to the Proof of Proposition 2.9 in Ref. 30.

Note that in the above proof, it is not important that v is precisely normal to $H1$⁠. Indeed, exactly the same proof allows us to obtain a similar conclusion for approximately null vectors as well.

Our next goal is to show that for α, β sufficiently small depending on K [indeed, the proof shows that we can take α < γ′/(K² log K) and β < γ′/K, where γ′ > 0 is a constant depending only on the distribution of the random variable ξ], we have

where γ ∈ (0, 1) is a constant depending only on the distribution of the random variable ξ.

For this, we begin with a standard decomposition of the unit sphere due to Rudelson and Vershynin.⁷³ 

We will also need the following lemma from Ref. 73.

Now, we are ready to prove (8). Consider a vector v ∈ Sⁿ⁻¹ such that $‖Xzv‖≤tbn−10mm−1/2$⁠, where 0 ≤ t ≤ 1. Then, on the event $EK$⁠, it follows from Proposition 2.7 that for any i ∈ [m],

Moreover, since for every i ∈ [m],

it follows that

Let $E$ denote the event $EK∩∩i∈[m]infw∈Compcn(a,κ)‖Miw‖>γ$⁠, where a, κ, and γ are as in Lemma 2.9. On the event $E$⁠, if t ≤ γ, then

Therefore, we can conclude from Lemma 2.10 that on the event $E$⁠, any vector v ∈ Sⁿ⁻¹ such that $‖Xzv‖≤γbn−10mm−1/2$ will have at least αn coordinates larger than $βbn−10mm−1/2bn−1/2$⁠, where α = γ′/(K² log K), β = γ′/K, and γ′ > 0 is a constant depending only on γ.

Hence, with this choice of α, β, γ′, the probability of the event in (8) is bounded by

where the last inequality follows by Lemma 2.9. This proves (8).

The next lemma is a direct consequence of Lemma 2.10 and Lemmas 2.5 and 2.7 from Ref. 32.