Extension of the Gaussian free field correlation theorem without ordering assumptions

Let $N=1,2,\dots$, let $\varkappa_j=(\nu_j,\eta_j,\tau_j)\in\mathcal D$ be distinct triples, and let $\Omega_j=\Omega(\nu_j,\eta_j,\tau_j)$. The correlation formula of Theorem 1.17 is the moment formula with covariance given by the Green function

$$\mathcal G(z,w)=-\frac{1}{2\pi}\ln\left|\frac{z-w}{z-\bar w}\right|$$

without imposing the ordering assumptions $\tau_1\leq\tau_2\leq\dots\leq\tau_N$ and $\eta_1\geq\eta_2\geq\dots\geq\eta_N$, provided that the $\Omega$-images of all the triples are pairwise distinct. Correlation theorem extension. The statement of Theorem 1.17 remains valid without the ordering assumption, provided that the points $\Omega_1,\dots,\Omega_N$ are pairwise distinct.