These numerics accompany this paper (arXiv:2207.09599) proving a probabilistic Weyl law for randomly perturbed Berezin—Toeplitz operators.

Vogel's theorem example

Here the eigenvalues of the Scottish flag operator with a random perturbation are computed for a fixed $N = 1000$. The number of eigenvalues is counted in a rectangular region, which closely matches the predicted Weyl-law proven by Martin Vogel (Link).

Analogue of Scottish Flag Operator

This is the spectrum of the Toeplitz quantization of a family of functions on $\mathbb C \mathbb P ^1$. Identifying $\mathbb C \mathbb P ^1$ with the sphere in $\mathbb R^3$ with coordinates $(x_1,x_2,x_3)$. The functions being quantized are $a (x_1^2 + x_1 ) + i x_2$ where $a$ goes between $0$ and $2$, and $N$ is fixed at $100$.

Probabilistic Weyl wlaw for $\mathbb{C} \mathbb{P}^1$

Here is the analogous numerical experiment as above. Here the eigenvalues of Toeplitz quantization of $x_1 + i x_2$ on the sphere, plus a small random perturbation, are computed. The number of eigenvalues within a circle of growing radius is computed, which closely matches a Weyl-law proven in (arXiv:2207.09599).

Spherical Stretching

This is the spectrum of the quantization of $a x_1^2 + x_1 + i x_2$, on $\mathbb C \mathbb P^1$ (with a random perturbation), where $a$ goes between $-1$ and $1$, and $N$ is fixed at $3000$.

Spherical Stretching Full Rotation

This is the spectrum of a Toeplitz quantization of $\sin(t) x_1^2 + \cos(t) x_1 + i x_2$ on $\mathbb C \mathbb P^1$ (with a random perturbation), where $t$ goes between $0$ and $ 2 \pi$, and $N$ is fixed at $3000$. This is an inadvertent example of the spinning dancer illusion.

Spherical Stretching Full Rotation Slowed Down

Same as above, but slowed down and for $N = 1000$. Note how the eigenvalues tend to repel each other.

Funny Numerical Noise

This is the same computation as above, but for a smaller range of $a$ and with no random perturbation added, any perturbation comes from numerical errors in Matlab.