Abstract:
We consider integer-valued functions on the square lattice which differ by exactly one between any two adjacent vertices. Such functions are known as height functions and the value at each vertex is called the height (though it can also be negative). For a given $V$ finite subset of vertices, we consider the set of all height functions that are equal to 0 (resp. 1) at each even (resp. odd) vertex in the complement of $V$. Given a positive parameter $c$, the probability distribution on this set of functions is defined to be proportional to $c$ to the number of pairs of diagonally adjacent sites having the same height. The central question is the order of fluctuations of the heights under this measure when the size of $V$ tends to infinity.
It turns out that the answer depends on the parameter $c$. We show that at $c=2$ the variance of the height at a given vertex diverges logarithmically in the distance between this vertex and the complement of $V$ (i.e. the height function is rough), while for any $c > 2$ the fluctuations are uniformly bounded (i.e. the height function is flat). This question is naturally linked to the presence (or absence) of the long-range order in the two classical models of statistical mechanics: six-vertex model and Ashkin-Teller model.
The central tools in the proofs are couplings between different models. In particular, we make use of a recent rigorous understanding of the order of the phase transition in the random-cluster model.
(joint work with Ron Peled)