We prove that the point process of the eigenvalues of real or complex non-Hermitian matrices $X$ with independent, identically distributed entries is hyperuniform: the variance of the number of eigenvalues in a subdomain $Ω$ of the spectrum is much smaller than the volume of $Ω$. Our main technical novelty is a very precise computation of the covariance between the resolvents of the Hermitization of $X-z_1, X-z_2$, for two distinct complex parameters $z_1,z_2$.
We consider the i.i.d. matrix ensemble, i.e. an ensemble of N × N non-Hermitian random matrices X with independent, identically distributed (i.i.d.) real or complex centered entries. For convenience, we normalize the entries of X so that E|X ij | 2 = N -1 . Then eigenvalues σ i of such matrices form a correlated point process on the unit disk of the complex plane; see [13,51] for the Ginibre ensemble 1 and [27,41,72,78] for general i.i.d. matrices (see also the recent [3,18,25,70,71,94]). These eigenvalues tend to be uniformly distributed over the unit disk D. In particular, for any (sufficiently nice) subdomain of the disk Ω ⊂ D, it is well known [7,52,54,80,87] that (circular law)
with high probability. In particular, this implies EN Ω ∼ N . Note that the lhs. of (1.1) is random, while the leading order term in the rhs. is deterministic. It is then natural to study the fluctuations around this deterministic limit. For a system of independent particles, such as a Poisson point process it holds that the number variance is proportional to the expectation, i.e. Var(N Ω ) ∼ EN Ω ∼ N . Thus, the natural question: Is the size of Var(N Ω ) for i.i.d. matrices still proportional to those of the mean? We answer it negatively; the number variance is much smaller than the expected size of N Ω :
Theorem 1.1 (Informal statement). Let X be an i.i.d. matrix and consider any nice domain Ω ⊂ D. Then, there exists a q > 0 such that
Our proof explicitly gives q = 1/40 in the complex case and q = 1/106 in the real case, see Theorems 2.4 and 2.6 below. Theorem 1.1 establishes a connection between the point process of the eigenvalues of general i.i.d. matrices with hyperuniformity, a key concept in condensed matter physics for classifying crystals, quasicrystals, and exotic states of matter. In [90, Section 1], Torquato defines the concept of hyperuniformity as: (…) the number variance of particles within a spherical observation window of radius R grows more slowly than the window volume in the large-R limit.
For a system of independent particles the number variance is proportional to the volume Var(N Ω ) ∼ N , hence it is not hyperuniform. Hyperuniformity is typically a signature of strong correlations at large distances in the system that reduce fluctuations. In random matrix theory, one well-known manifestation of such correlations is the eigenvalue rigidity, proven in many Hermitian random matrix models, which asserts that each eigenvalue fluctuates on a scale only slightly larger (by an N ϵ factor) than the typical distance between neighbouring eigenvalues. In particular, this trivially implies hyperuniformity in the sense of Torquato. In fact, for Hermitian models with eigenvalue rigidity, the variance of N I , the number of eigenvalues within an interval I ⊂ R, is bounded by N ϵ , independently of I. Note that here the concept of rigidity uses that the Hermitian spectrum is one dimensional hence the eigenvalues can be ordered. For the same reason, the proof of eigenvalue rigidity easily follows from optimal concentration properties of the resolvent (optimal local laws). In the two-dimensional setting of non-Hermitian random matrices, there is no direct concept of eigenvalue rigidity, but hyperuniformity can be interpreted as a higher dimensional version of the same strong correlations that cause rigidity. However, in this case hyperuniformity does not simply follow from optimal local laws (which are well understood in this setting as well); substantial new inputs are needed.
The word hyperuniformity was coined by Torquato and Stillinger [92] in the early 2000, analogous concepts appeared in the physics of Coulomb gases much earlier [60,66,68,74,75]. In recent years this phenomenon attracted lots of interest in a variety of models in mathematics and physics (here for concreteness we focus only on two-dimensional objects), including averaged and perturbed lattices [50], zeroes of random polynomials with i.i.d. Gaussian coefficients [48,85], certain fermionic systems [17,91], Berezin-Toeplitz operators in connection with the Quantum Hall Effect [19]. A rigorous proof of hyperuniformity in system with two-body interactions is notoriously difficult. In the prominent model of the two-dimensional Coulomb gas (also known as the two-dimensional one component plasma) hyperuniformity was proved only very recently by Leblé [65]. This result showed that the number variance is bounded by N/(log N ) a , for some small a > 0; so, in particular, it grows more slowly than the volume N .
Our result (1.2) shows that the point process given by the eigenvalues of general non-Hermitian i.i.d. matrices is hyperuniform. It can thus be thought as a random matrix counterpart of [65], with a much stronger control on the number variance: we prove a polynomial factor N q , while in the Coulomb gas model only a logarithmic factor (log N ) a was obtained. Previous to our result, in the random matrix setting, hyperuniformity was known only for i
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