Pointwise Weyl Law for Graphs from Quantized Interval Maps

Abstract

We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pakoński et al. (J Phys A 34:9303–9317, 2001) as a model for quantum chaos on graphs. Since we allow shrinking spectral windows in the pointwise Weyl law, as a consequence we obtain for these models a strengthening of the quantum ergodic theorem from Berkolaiko et al. (Commun Math Phys 273:137–159, 2007), and show in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. We also examine further the specific case where the interval map is the doubling map.

Appendix A. Details for Sect. 6

In this section we provide details concerning Theorem 6.1. The following theorem from [36], also using [16], is a quantitative version of the theorem from [18]. Theorem 6.1 will follow from this theorem applied to our case with complex projections.

Theorem A.1

(Complex version of Theorem 2 in [36]). Let $\{x_j\}_{j=1}^n$ be deterministic vectors in $\mathbb {C}^d$, normalized so that $\sigma ^2=\frac{1}{nd}\sum _{i=1}^n\Vert x_i\Vert ^2=1$ and suppose

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n\left| \Vert x_i\Vert ^2-d\right|&\le A \end{aligned}$$

(A.1)

$$\begin{aligned} \sup _{\theta \in \mathbb {S}_\mathbb {C}^{d-1}}\frac{1}{n}\sum _{i=1}^n|\langle \theta ,x_i\rangle |^2&\le B. \end{aligned}$$

(A.2)

For a point $\theta \in \mathbb {S}_\mathbb {C}^{d-1}\subset \mathbb {C}^d$, define the measure $\mu ^{(n)}_\theta :=\frac{1}{n}\sum _{j=1}^n\delta _{\langle \theta ,x_j\rangle }$ on $\mathbb {C}$. There are absolute numerical constants $C,c>0$ so that for $\theta \sim {\text {Unif}}(\mathbb {S}_{\mathbb {C}^{d-1}})$, any bounded L-Lipschitz $f:\mathbb {C}\rightarrow \mathbb {C}$, and $\varepsilon >\frac{2L(A+3)}{d-1}$, there is the quantitative bound

$$\begin{aligned} \mathbb {P}\left[ \left| \int f(x)\,\textrm{d}\mu ^{(n)}_\theta (x)-\mathbb {E}f( Z)\right| >\varepsilon \right] \le C\exp \Big (-\frac{c\varepsilon ^2d}{L^2B}\Big ), \end{aligned}$$

(A.3)

where $Z\sim N_\mathbb {C}(0,1)$. In particular, if $d=d(n)\rightarrow \infty$ as $n\rightarrow \infty$, and $A=o(d)$ and $B=o(d)$, then $\mu ^{(n)}$ converges weakly in probability to $N_\mathbb {C}(0,1)$ as $n\rightarrow \infty$.

Proof outline of Theorem A.1

The proof is the same as the real version in [36], except that the (multi-dimensional) Theorem A.2 written below from [16] replaces the single-variable version. The proof idea from [36] is to let $F(\theta ):=\frac{1}{n}\sum _{i=1}^nf(\langle \theta ,x_i\rangle )$ and write

$$\begin{aligned} \mathbb {P}\left[ \left| F(\theta )-\mathbb {E}f( Z)\right|>\varepsilon \right]&\le \mathbb {P}\left[ \left| F(\theta )-\mathbb {E}F(\theta )\right| >\varepsilon -|\mathbb {E}F(\theta )-\mathbb {E}f(Z)|\right] . \end{aligned}$$

Then one uses Theorem A.2, a generalization of Stein’s method of exchangeable pairs for abstract normal approximation, to bound $|\mathbb {E}F(\theta )-\mathbb {E}f(Z)|$ with $W=\langle \theta ,x_I\rangle$ where $I\sim {\text {Unif}}\llbracket n\rrbracket$, and then one can apply Gaussian concentration (Lemma A.3, see for example [12, §5.4]) to F which is $(L\sqrt{B})$-Lipschitz. $\square$

In the following, $\mathcal {L}(X)$ will denote the law of a random variable or vector X.

Theorem A.2

(Theorem 2.5 for $\mathbb {C}$ in [16]). Let W be a $\mathbb {C}$-valued random variable and for each $\varepsilon >0$ let $W_\varepsilon$ be a random variable such that $\mathcal {L}(W)=\mathcal {L}(W_\varepsilon )$, with the property that $\lim _{\varepsilon \rightarrow 0}W_\varepsilon =W$ almost surely. Suppose there is a function $\lambda (\varepsilon )$ and complex random variables $\Gamma ,\Lambda$ such that as $\varepsilon \rightarrow 0$,

• (i) $\frac{1}{\lambda (\varepsilon )}\mathbb {E}[(W_\varepsilon -W)|W]\xrightarrow {L^1}-W$.

• (ii) $\frac{1}{2\lambda (\varepsilon )}\mathbb {E}[(W_\varepsilon -W|^2|W]\xrightarrow {L^1}1+\mathbb {E}[\Gamma |W]$.

• (iii) $\frac{1}{2\lambda (\varepsilon )}\mathbb {E}[(W_\varepsilon -W)^2|W]\xrightarrow {L^1}\mathbb {E}[\Lambda |W]$.

• (iv) $\frac{1}{\lambda (\varepsilon )}\mathbb {E}|W_\varepsilon -W|^3\rightarrow 0$.

Then letting $Z\sim N_\mathbb {C}(0,1)$,

$$\begin{aligned} d_\textrm{Wass}(W,Z)\le \mathbb {E}|\Gamma |+\mathbb {E}|\Lambda |, \end{aligned}$$

(A.4)

where $d_\textrm{Wass}$ is the Wasserstein distance $d_\textrm{Wass}(W,Z)=\sup _{\Vert g\Vert _\textrm{Lip}\le 1}|\mathbb {E}g(W)-\mathbb {E}g(Z)|$.

Lemma A.3

(Gaussian concentration on the complex sphere). Let $F:\mathbb {C}^d\rightarrow \mathbb {C}$ be L-Lipschitz and $\theta \sim {\text {Unif}}(\mathbb {S}_\mathbb {C}^{d-1})$. Then there are absolute constants $C,c>0$ so that

$$\begin{aligned} \mathbb {P}[|F(\theta )-\mathbb {E}F(\theta )|\ge t]\le C\exp (-ct^2d/L^2). \end{aligned}$$

(A.5)

1.1 A.1. Proof of Theorem 6.1

Let $v_1,\ldots ,v_d$ be an orthonormal basis for $V^{(\nu )}$, and let $M_V$ be the $n\times d$ matrix with those vectors as columns. Apply Theorem A.1 to the data set $\sqrt{n}M_V^*e_1,\sqrt{n}M_V^*e_2, \ldots ,\sqrt{n}M_V^*e_n$ in $\mathbb {C}^d$, noting that since $P^{(\nu )}=M_VM_V^*$, then

$$\begin{aligned} \sigma ^2=\frac{1}{nd}\sum _{x=1}^nn\Vert M_V^*e_x\Vert _2^2=\frac{1}{d}\sum _{x=1}^N\langle e_x|P^{(\nu )}|e_x\rangle =\frac{1}{d}{\text {tr}}P^{(\nu )}=1. \end{aligned}$$

We can also take $B=1$ since for any $\theta \in \mathbb {S}_\mathbb {C}^{d-1}$,

$$\begin{aligned} \frac{1}{n}\sum _{x=1}^n|\langle \theta ,\sqrt{n}M_V^*e_x\rangle |^2=\sum _{x=1}^n\langle M_V\theta ,e_x\rangle \langle e_x,M_V\theta \rangle =\Vert M_V\theta \Vert _{\mathbb {C}^n}^2= \Vert \theta \Vert _{\mathbb {C}^d}^2. \end{aligned}$$

If $\theta$ is uniform on $\mathbb {S}_\mathbb {C}^{d-1}\subset \mathbb {C}^d$, then $M_V\theta$ is uniform on $\mathbb {S}(V^{(\nu )})$, so $\frac{1}{n}\sum _{j=1}^n\delta _{\langle \theta ,\sqrt{n}M_V^*e_x\rangle }\sim \frac{1}{n}\sum _{j=1}^n\delta _{\sqrt{n}\omega _x}$.