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Nesting Field of Double-Dimer Loops and Convergence to CLE(4)
Establishes that the normalized fluctuations of the number of double-dimer loops surrounding a fixed point in the discretized upper half-plane are asymptotically Gaussian, and that the double-dimer nesting field converges in distribution, in the local Sobolev space H^{-1-nu}, to the nesting field of the conformal loop ensemble CLE(4). The proof controls the near-diagonal asymptotics of the inverse Kasteleyn operator carrying an SL(2,C) monodromy and links it, via an approximate Wick identity for the height function, to the Laplace transform of the loop-counting variable.