Consider a Gibbs random field \(X= (X_i)_{1\leq i\leq N}\) with \(X_i\in\mathbb{N}\) for all \(i\) and effective Hamiltonian \[ H_\lambda= \sum^{N-1}_{i=1} \Phi(X_{i+1}- X_i)+ \sum^N_{i=1} V_\lambda(X_i) \] depending on a parameter \(\lambda>0\). For fairly general interactions \(\Phi\) and external potentials \(V_\lambda\), the authors derive scaling limits for \(X\), letting \(N\to\infty\) and \(\lambda(N)\to 0\) in a suitably connected way. The limiting objects are stationary, reversible, ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated Sturm-Liouville operators.
In particular, if \(V_\lambda= x\lambda\), the Ferrari-Spohn diffusion with log-Airy drift is recovered [P. L. Ferrari and H. Spohn, Ann. Probab. 33, No. 4, 1302–1325 (2005; Zbl 1082.60071)]. The proofs combine functional analysis with probabilistic estimates for random walks based on an earlier paper [O. Hryniv and Y. Velenik, Probab. Theory Relat. Fields 130, No. 2, 222–258 (2004; Zbl 1078.60034)]. Some applications to problems of statistical physics are mentioned, such as critical prewetting in the two-dimensional Ising model, interfacial adsorption between equilibrium phases, geometry of the top-most layer of the \(2+1\)-dimensional SOS model above a wall, and island of activity in kinetically constrained models.