Nesting of double-dimer loops: local fluctuations and convergence to the nesting field of CLE(4)

We consider the double-dimer model in the upper-half plane discretized by the square lattice with mesh size $\delta$. For each point $x$ in the upper half-plane, we consider the random variable $N_{\delta }(x)$ given by the number of the double-dimer loops surrounding this point. We prove that the normalized fluctuations of $N_{\delta }(x)$ for a fixed $x$ are asymptotically Gaussian as $\delta \to 0+$. Further, we prove that the double-dimer nesting field $N_{\delta }(\cdot )-\mathbb{E}{N}_{\delta }(\cdot )$, viewed as a random distribution in the upper half-plane, converges as $\delta \to 0+$ to the nesting field of CLE(4) constructed by Miller, Watson and Wilson.