Fractional-statistics emergence conjecture for quasihole tracer particles

Fix an integer $n\ge 2$. Let $p,\mu\in\mathbb{N}$, let $\mathfrak{b}=N\in\mathbb{N}$, and define

$$\Psi_{\rm qh}(\mathbf{w};\mathbf{z})=\left(\prod_{j=1}^n\prod_{k=1}^N(w_j-z_k)^p\right)\left(\prod_{1\leq k<\ell\leq N}(z_k-z_\ell)^\mu\prod_{k=1}^N e^{-\mathfrak{b}|z_k|^2/2}\right).$$

Let $\Phi\in C^0(\mathbb{R}^{2n})$ have support in $\{ |x|\leq d\}^{\times n}$ for some $d<\sqrt{\mu}$, and assume that for some $C_\Phi>0$ and $\beta>0$,

$$|\Phi(\mathbf{y})|\leq C_\Phi |y_i-y_j|^\beta$$

for all $i\ne j$. For any $j\in\{1,\dots,n\}$, and for sufficiently large $\kappa>0$, define

$$\mathbf{A}^{\rm tot}_j(\mathbf{y})=-\left(q-\frac{p}{\mu}\right)\mathfrak{b}y_j^\perp-\frac{p^2}{\mu}\sum_{\ell\ne j}\frac{(y_j-y_\ell)^\perp}{|y_j-y_\ell|^2}.$$

Fractional-statistics emergence conjecture. As $N\to+\infty$, the identity

$$\int_{\mathbb{R}^{2(N+n)}}\left|\left(-\mathrm{i}\nabla_{y_j}-q\mathfrak{b}y_j^\perp\right)\Psi_\Phi\right|^2=2\mathfrak{b}\frac{p}{\mu}+\int_{\mathbb{R}^{2n}}\left|\left(-\mathrm{i}\nabla_{y_j}+\mathbf{A}^{\rm tot}_j(\mathbf{y})\right)\Phi\right|^2+\mathrm{Errors}$$

should hold. This conjecture extends the paper's statistics-transmutation result from integer to fractional statistics, motivated by the fractional quantum Hall effect; the notation $\Psi_\Phi$ is inherited from the preceding definition in the paper, and the source does not state a quantified form of the error term.