A Probabilistic Weyl-Law for Perturbed Berezin-Toeplitz Operators

This paper generalizes a result of Martin Vogel in [23] which proves a probabilistic Weyl-law for quantizations of functions on tori. Here we do the same, but with the tori replaced by arbitrary Kähler manifolds equipped with positive line bundles.

In [23], Vogel considers Toeplitz quantizations of smooth functions on a real $2d$-dimensional torus, which associates every smooth function $f$ on the torus to a family of $N^{d}\times N^{d}$ matrices, $f_{N}$, for all $N\in\mathbb{N}$ (here $N^{-1}$ is the semi-classical parameter). A recent physical motivation for such constructions is written by Deleporte in [6, Section 1]. Next, a random matrix with sufficiently small norm is added to $f_{N}$, and the spectrum is shown to obey an almost-sure Weyl-law as $N$ goes to infinity. This was conjectured by Christiansen and Zworski in [4] and is a major extension of their work.

This result is most striking when the unperturbed matrix is non-self-adjoint. For example, if $f(x)=\cos(2\pi x)+i\cos(2\pi\xi)$, then the quantization is

which numerically has spectrum contained on two crossing lines in the complex plane. This operator is aptly named the Scottish flag operator and is further described by Embree and Trefethen in [9]. Interestingly, (as far as we are aware) it is unknown analytically where the spectrum of $f_{N}$ lives. However, if randomly perturbed, the spectrum spreads out with density given by the push-forward of the Lebesgue measure on the torus by $f$. Figure 1 plots the spectrum of $f_{N}$ with no perturbation, and with a small perturbation.

The spectral properties of randomly perturbed non-self-adjoint operators was pioneered by Hager in [11], in which the operator $hD_{x}+g(x):L^{2}(S^{1})\to L^{2}(S^{1})$ was studied. This result, and numerous subsequent results are discussed by Sjöstrand in [16]. There are related results describing spectral properties of randomly perturbed Toeplitz matrices, which can be defined as quantizations of symbols on $\mathbb{T}^{2}$ with symbol independent of $x$. See Davies and Hager [5], Guionnet, Wood and Zeitouni [10], Sjöstrand and Vogel [18] [17], and references given there.

This paper is the natural generalization of Vogel’s result in [23]. Here we prove a similar result for quantizations of functions on Kähler manifolds (with sufficient structure, as discussed in Section 2). These quantizations, called Berezin-Toeplitz operators (or just Toeplitz operators) were first described by Berezin in [2] as a particular type of quantization of symplectic manifolds. Following [2], for every smooth function $f$ on a quantizable Kähler manifold $X$, we get a family of finite rank operators, $T_{N}f$, indexed by $N\in\mathbb{N}$ (see [14] for a connection between these quantizations, and quantizations on the torus) which have physical interpretations. Deleporte in [6, Appendix A] relates this quantization to spin systems in the large spin limit, and Douglas and Klevtsov in [7] use path integrals for particles in a magnetic field to derive the Bergman kernel (a key ingredient in constructing $T_{N}f$).

Next, if we add a small Gaussian-type random perturbation $\mathcal{G}_{\omega}$ to these operators (see Definition 2.3), the empirical measures weakly converge almost surely (see Theorem 2 in Section 2 for a precise statement). Theorem 3 states a result about more general random perturbations $\mathcal{W}_{\omega}$ (see Definition 2.3) but with a more restrictive coupling constant. A consequence of Theorem 3 is the following probabilistic Weyl-law.

Finer results are expected for describing the spectrum of randomly perturbed Toeplitz operators. In [23], precise statements about the number of eigenvalues are obtained using counting functions of holomorphic functions. Here we only show weak convergence of the empirical measures, but achieve this in a relatively simple way using logarithmic potentials as presented in [20].

Here we present numerical examples to motivate the main result of this paper. Consider the Kähler manifold $\mathbb{C}\mathbb{P}^{1}$ (complex protective space of dimension $1$) which can be identified with the real $2$-sphere with coordinates $(x_{1},x_{2},x_{3})$. In Figure 2, we compute the spectrum of the quantization of the function $f=x_{1}+2x_{2}^{2}+ix_{2}$. Before perturbation, the spectrum lies on several lines in the complex plane, somewhat analogous to the Scottish flag operator. However, as a perturbation is added, the spectrum fills in. This paper describes the structure of the spectrum of this perturbed operator in the semiclassical limit, as $N\to\infty$.

Numerical verification of this paper’s result can be seen if $f=ix_{1}+x_{2}$ (still on $\mathbb{C}\mathbb{P}^{1}$). Figure 3 computes the spectrum of $T_{N}f$ with a random perturbation added, and plots the number of eigenvalues in circles of increasing radii versus the predicted number of such eigenvalues by Theorem 1. More animations can be found on my website¹¹1 https://math.berkeley.edu/~izak/research/toeplitz/movies.html.

Let $(X,\sigma)$ be a compact, connected, $d-$dimensional Kähler manifold with a holomorphic line bundle $L$ with positively curved Hermitian metric locally given by $h=e^{-\varphi}$. That is over each fiber $x\in X$, $\left\|v\right\|_{h}:=e^{-\varphi(x)}|v|$. Given this, the globally defined symplectic form, $\sigma$, is related to the Hermitian metric by $i\partial\overline{\partial}\varphi=\sigma$. Fixing local trivializations, $\varphi$ can be described as a strictly plurisubharmonic smooth real-valued function (called the Kähler potential). This is further outlined by Le Floch in [12].

Let $L^{N}$ be the $N$th tensor power of $L$, which has Hermitian metric $h_{N}:=e^{-N\varphi}$. Let $\mu_{d}=\sigma^{\land d}/d!$ be the Liouville volume form on $X$. This provides an $L^{2}$ structure on sections of $L^{N}$. Indeed, if $u$ and $v$ are smooth sections on $L^{N}$, then define

Define $L^{2}(X,L^{N})$ to be the space of smooth sections of $L^{N}$ with finite $L^{2}$ norm. In this $L^{2}$ space, let $H^{0}(X,L^{N})$ be the space of holomorphic sections.

For the remainder of this paper, denote $\dim(H^{0},(X,L^{N}))$ by $\mathcal{N}=\mathcal{N}(N)$. The orthogonal projection from $L^{2}(X,L^{N})$ to $H^{0}(X,L^{N})$ is called the Bergman projector and is denoted by $\Pi_{N}$. Finally, given $f\in C^{\infty}(X;\mathbb{C})$, the Toeplitz operators associated to $f$, written $T_{N}f$, are defined for each $N\in\mathbb{N}$ as $T_{N}f(u)=\Pi_{N}(fu)$, where $u\in H^{0}(X,L^{N})$. In this way, $T_{N}f$ are finite rank operators mapping $H^{0}(X,L^{N})$ to itself. For the remainder of this paper, we will fix a basis for $H^{0}(X,L^{N})$ so that $T_{N}f$ (and similar operators) can be considered as matrices.

The class of functions to quantize will often depend on $N$. To define this symbol class requires local control of functions. Fix a finite atlas of neighborhoods $(U_{i},\zeta_{i})_{i\in\mathcal{I}}$ for the Kähler manifold $X$.

If $f\sim\sum N^{-j}f_{j}$, we call $f_{0}$ the principal symbol of $f$, which is unique modulo $\mathcal{O}(N^{-1})$.

We next add a random perturbation to these Toeplitz operators. For this we must fix a probability space $\Omega$ with probability measure $\mathbb{P}$.

The $\omega$ in the subscript of these objects is to emphasize that these objects are random. That is for each $\omega\in\Omega$, $\mathcal{G}_{\omega}$ is a finite rank operator. The majority of this article describes perturbations by $\mathcal{G}_{\omega}$ (the Gaussian case), while a brief note at the end concerns the more general perturbations by $\mathcal{\textbullet W}_{\omega}$.

This paper will prove almost sure weak convergence of the empirical distribution of eigenvalues of randomly perturbed Toeplitz operators. The principal symbol of $f$ must also satisfy the property that there exists $\kappa\in(0,1]$ such that

as $t\to 0$ uniformly for all $z\in\mathbb{C}$. It is observed in [4] that if $f$ is real analytic, then (2.5) holds. See [4], and references presented there, for further discussion of (2.5).

We expect Theorem 2 to hold for a much larger class of random perturbations than described in Definition 2.3. Indeed, the only properties of $\mathcal{G}_{\omega}$ we use is a norm bound (Lemma 4.6) and an anti-concentration bound (Proposition 5.7). See [24] where Vogel and Zeitouni establish similar logarithmic determinant estimates with these classes of random perturbations, and [1, Remark 1.3] where Basak, Paquette, and Zeitouni describe random perturbations satisfying these properties.

Here we present a version of Theorem 2 for the more general random perturbations $\mathcal{W}_{\omega}$ as described in Definition 2.3.

A proof of this result is presented in Section 8.

In proving Theorem 2, non-negative symbols are scaled by powers of $N^{-1}$. These functions belong to a more exotic symbol class than smooth functions uniformly bounded in $N$. Toeplitz operators of functions in this symbol class still have natural composition formulas. A summary of these results is contained in this section. For proofs see [13].

Typically, Proposition 3.5 will be applied with $R_{N}=0$ for all $N$.

Note that if $f=1$, then $\operatorname{Tr}T_{N}1=\operatorname{Tr}(\Pi_{N})=\dim(H^{0}(X,L^{N}))=\mathcal{N}$ which is an alternative way of proving that $\mathcal{N}=\operatorname{vol}(X)(N/2\pi)^{d}+\mathcal{O}(N^{d-1})$.

This paper uses the probabilistic machinery of logarithmic potentials. A brief overview is presented in this section.

Using the fact that $\log|z|$ is the fundamental solution of the Laplacian, it can be shown that, in the sense of distributions, $\Delta U_{\nu}=2\pi\nu$, which is the key ingredient in proving the following theorem.

We wish to use Proposition 4.3 to prove almost sure weak convergence of the empirical measures of $T_{N}f+\delta\mathcal{G}_{\omega}$.

Where $\strokedint_{X}f\mathop{}\!\mathrm{d}\mu_{d}$ is defined as $\operatorname{vol}(X)^{-1}\int f\mathop{}\!\mathrm{d}\mu_{d}$.

Let $\Lambda$ be a neighborhood of $f(X)$. Clearly $\operatorname{supp}\nu\subset\Lambda$, the same is true with probability $1$ for $\nu_{N}$, for sufficiently large $N$. A standard random matrix lemma is required to show this.

For a fixed $\varepsilon>0$, we will choose $\delta=\delta(N)$ such that

Therefore $\nu_{N}$ and $\nu$ satisfy the conditions of Proposition 4.3. So it suffices to show that $U_{\nu_{N}}(z)\to U_{\nu}(z)$ for almost all $z$ in the bounded set containing $\Lambda$. To prove this almost sure convergence, it suffices to apply Lemma 4.9 with $Y_{N}=\mathcal{N}^{-1}\log|\det(T_{N}f+\delta\mathcal{G}_{\omega}-z)|$ and $Y=\strokedint\log|z-f_{0}(x)|\mathop{}\!\mathrm{d}\mu_{d}(x)$ for suitably chosen $\varepsilon_{N}$.

To control $\log|\det(T_{N}f+\delta\mathcal{G}_{\omega}-z)|$ we follow the now standard method of setting up a Grushin problem. This approach was used in [23] and [11], and is comprehensively reviewed in [19].

Let $P=T_{N}f$ and $\mathcal{H}_{N}=H^{0}(X,L^{N})$. Define the $z$-dependent self-adjoint operators $Q=(P-z)^{*}(P-z)$ and $\tilde{Q}=(P-z)(P-z)^{*}$. These operators share the same eigenvalues $0\leq t_{1}^{2}\leq\cdots\leq t_{\mathcal{N}}^{2}$. We can find an orthonormal basis of eigenvectors of $Q$ for these eigenvalues, denoted by $e_{i}$, and similarly, and orthonormal basis of eigenvectors of $\tilde{Q}$ denoted by $f_{i}$. These eigenvectors can be chosen such that

Next we fix $\rho\in(0,\min(1/2,\varepsilon))$, and define:

To ease notation, the $z$ in the argument for these operators will often be dropped. Unless specified, all estimates are uniform in $z$.

These lemmas, along with Lemma 4.6, guarantee that if $\delta=\mathcal{O}(\alpha^{1/2}\mathcal{N}^{-1/2})$, then $\mathcal{P}^{\delta}$ is invertible with overwhelming probability. Denote the inverse of $\mathcal{P}^{\delta}$ by $\mathcal{E}^{\delta}$ with the same notation for its components as in (5.4).

Define $P^{\delta}=P+\delta\mathcal{G}_{\omega}$. By Schur’s complement formula, if $P^{\delta}-z$ is invertible,

Writing $\mathcal{P}^{\delta}\mathcal{E}^{\delta}=1$, we get that $-R_{-}=(P^{\delta}-z)E_{+}^{\delta}(E_{-+}^{\delta})^{-1}$ and $R_{+}E_{+}^{\delta}=1$. Therefore $-R_{+}(P^{\delta}-z)^{-1}R_{-}=(E_{-+}^{\delta})^{-1}$, so that

Note that $P^{\delta}-z$ is invertible if and only if $E_{-+}^{\delta}$ is invertible. Therefore (5.7) holds even when $P^{\delta}-z$ is not invertible.

Therefore, to prove Theorem 2, it suffices to show summability of the probability of the events:

We let $\varepsilon_{N}=N^{-\gamma}$ for a suitably chosen $\gamma=\gamma(d,\kappa)>0$. Expand $B=B_{1}+B_{2}+B_{3}$ where:

Controlling $B_{1}$ requires the most work as it requires utilizing the calculus of Toeplitz operators. However, it is completely deterministic, and remains true for unperturbed operators. $B_{2}$ will be easily shown to be negligible. Proving a lower bound on $B_{3}$ is the key ingredient in proving Theorem 2, as it will force the events $\mathcal{A}_{N}$ to sufficiently small probability. Without a perturbation, $B_{3}$ will have no lower bound.