Gaussian free field limits throughout the balanced six-vertex regime

For every fixed c\in(0,2], the prescribed iterated balanced-torus limit \P_c exists. Define

As \delta\downarrow0 through arbitrary positive real values, the fields h_\delta converge to \sigma(c)\Gamma in each of the following senses.

1. 1.

   For every finite family \mu_1,\ldots,\mu_m of compactly supported finite signed measures of total mass zero and finite energy,

   \bigl(\langle h_\delta,\mu_j\rangle\bigr)_{j=1}^m \ \Longrightarrow\ \mathcal N\!\left(0, \bigl(\sigma(c)^2\mathcal E(\mu_i,\mu_j)\bigr)_{i,j=1}^m\right).
2. 2.

   For every bounded open U\subset\R^2 and -1<\alpha<0, there is convergence in law, with tightness, in the ordinary restriction spaces

   \mathcal C^\alpha_{\mathrm{res}}(U):=B^\alpha_{\infty,\infty}(U), \qquad B^\alpha_{p,q}(U),\qquad W^{\alpha,p}(U), \quad 1\le p<\infty,\quad 1\le q\le\infty,

   modulo constants. Convergence also holds in the outward completion (1.4) defined below. No regularity of \partial U is required. One may fix a smooth compactly supported \rho of integral one and replace each field X by X-\langle X,\rho\rangle, using the same pinning for the limit.

3. 3.

   For every k\ge1 and every tuple (u_i,u_i')_{i=1}^k with \{u_i,u_i'\}\cap\{u_j,u_j'\}=\varnothing for i\ne j,

   \E_c\prod_{i=1}^k \bigl(h_\delta(u_i')-h_\delta(u_i)\bigr) \longrightarrow \sigma(c)^k\sum_{\pi}\prod_{\{i,j\}\in\pi}C_{ij},

   where the sum is over pairings of \{1,\ldots,k\}, is zero for odd k, and

   C_{ij}=G(u_i',u_j')-G(u_i',u_j)-G(u_i,u_j')+G(u_i,u_j).

   The convergence is uniform on compact sets of such tuples.