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

TL;DR

This paper proves that the fluctuations of double-dimer loop counts in a discretized upper-half plane become Gaussian as the mesh size shrinks and that the nesting field converges to the CLE(4) nesting field, linking discrete models to conformal loop ensembles.

Contribution

It establishes the Gaussian nature of fluctuations and the convergence of the double-dimer nesting field to CLE(4), providing a rigorous connection between discrete models and conformal loop ensembles.

Findings

Normalized fluctuations are asymptotically Gaussian.

Nesting field converges to CLE(4) nesting field.

Results link discrete double-dimer models to conformal invariance.

Abstract

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.