This is a translation into math of the following question, posted on PhysicsOverflow.
Let $H:=L^2(\mathbb C)$.
For every $N$, let $\psi_N\in\Lambda^N H\cong (L^2(\mathbb C^N))^{S_N}$ be the function
$$
\psi_N(z_1,\ldots,z_N):=c\cdot\prod_{i<j}(z_i-z_j)^3\prod_i e^{-|z_i|^2/4},
$$
normalized so as to have $\|\psi_N\|^2=1$.
(Note that I could have replaced the $3$ above by any other odd integer — those are also called `Laughlin wave functions'.)
Let $D\subset \mathbb C$ be a unit disc, not necessarily centered at the origin. Let $H_{in}=L^2(D)$ and $H_{out}=L^2(\mathbb C\setminus D)$, so that $H=H_{in}\oplus H_{out}$. We then have a decomposition $$ \Lambda^N H=\bigoplus_{k=0}^N (\Lambda^kH_{in})\otimes (\Lambda^{N-k}H_{out}) $$ Write $\psi_{N,k}$ for the $k$-th summand of $\psi_N$ in the above direct sum decomposition.
Question 1: For any given $k$, is the limit $$ \lim_{N\to \infty} \|\psi_{N,k}\|^2 $$ (assuming it exists) independent of $D$?
Definition: Given a unit vector $\zeta=\sum_i\xi_i\otimes \eta_i\in H\otimes K$ in the tensor product of two Hilbert spaces, the `reduced density matrix' is the operator $[\zeta]_H:H\to H$ given by $$[\zeta]_H(x):= \sum_i \langle x,\xi_i\rangle \|\eta_i\|^2 \xi_i.$$ Equivalently, $[\zeta]_H$ is the partial trace of the rank one projection onto the line spanned by $\zeta$. (Sub-question: what's the correct math terminology and notation for $[\zeta]_H$?)
Let us from now on abbreviate $[-]_{H_{in}}$ by $[-]_{in}$.
Question 2 (which implies Question 1): For any given $k$, is the limit $$ \lim_{N\to \infty} [\psi_{N,k}]_{in} $$ (assuming it exists) independent of $D$?
Here, to make sense of Question 2, we need to identify the Hilbert spaces $H_{in}=L^2(D_1)$ and $H_{in}=L^2(D_2)$ associated to any two discs $D_1,D_2\subset \mathbb C$. This is done by the `magnetic translation' operator, which sends $f\in L^2(D_1)$ to $f_\tau\in L^2(D_2)$ defined by $$f_\tau(z):=e^{\mathrm{Im}(\bar \tau z)}f(z-\tau)$$ where $D_2=D_1+\tau$. (and I may be missing some minus sign or $2\pi$ in the exponent.)