math.PR — Probability

Closing the identification step in the GFF limit: Uniqueness of the zero-slope six-vertex phase for −1 ≤ Δ < 1/2

Contributed by Yan Ru Pei

For the six-vertex model with $a=b=1$ and $c\in(1,2]$, that is $-1\le\Delta<\frac{1}{2}$, we prove that the plane measure, the limit of balanced torus measures, is the only translation-ergodic gradient Gibbs measure of slope zero, and hence the unique translation-invariant minimiser of the specific free energy. Both statements had been asserted by reference to Sheffield's uniqueness theory, which is proved for nearest-neighbour potentials; we show that for $c\ne1$ the six-vertex weights, which couple faces across the diagonals of each vertex, are not a nearest-neighbour potential in height coordinates. The proof extends Sheffield's cluster swapping to these diagonal interactions by means of residual energies on the diagonals, shows with the Lanford–Ruelle theorem for subshifts of finite type and a Burton–Keane argument that two distinct ergodic zero-slope phases are gradients of Gibbs measures on heights, and shows from the spin representation in the form given by Duminil-Copin, Kozlowski, Lammers and Manolescu that the plane measure is not. This supplies a step that we found missing in the chain of recent results on the Gaussian free field limit of the six-vertex height function. An unrefereed manuscript of OpenAI states the full-plane limit for every $c\in(0,2]$; for $\sqrt{2}\le c<2$ it identifies a state built from nearly balanced sectors with the plane measure through the characterisation above. Our result gives that identification, given two properties of the state stated in the manuscript. We also check that the manuscript and the work of Averous, Duminil-Copin, He, Lammers and Manolescu concern the same measure, so that their convergence in domains holds, for $c\in[1,2]$ and with constants depending on $\Delta$, wherever the manuscript's full-plane theorem holds.