On intermediate level sets of two-dimensional discrete Gaussian free field

We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains D ⊂ C and describe the scaling limit, including local structure, of the level sets at heights growing as a λ-multiple of the height of the absolute maximum, for any λ ∈ (0, 1). We prove that, in the scaling limit, the scaled spatial position of a typical point x sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in D at parameter equal λ-times its critical value, the field value at x has an exponential intensity measure and the configuration near x reduced by the value at x has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges to that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by Daviaud (Ann. Probab. 34 (2006) 962–986).