Phase coexistence of gradient Gibbs states

Abstract

We consider the (scalar) gradient fields η a = (η b )—with b denoting the nearest-neighbor edges in \(\mathbb{Z}^{2}\)—that are distributed according to the Gibbs measure proportional to \(\hbox{e}^{-\beta H(\eta)}\nu(d \eta)\). Here H = ∑ b V(η b ) is the Hamiltonian, V is a symmetric potential, β > 0 is the inverse temperature, and ν is the Lebesgue measure on the linear space defined by imposing the loop condition \(\eta_{b_1}+\eta_{b_2}=\eta_{b_3}+\eta_{b_4}\) for each plaquette (b 1,b 2,b 3,b 4) in \(\mathbb{Z}^{2}\). For convex V, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex V undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature β. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., E η b = 0.