Solid-On-Solid interfaces with disordered pinning

We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $\phi: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(\omega_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $\alpha,\beta>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expression $$\mathcal H(\phi):= \beta\sum_{x\sim y} |\phi(x)-\phi(y)|- \sum_{x} (\alpha\omega_x+h){\bf 1}_{\{\phi(x)=0\}}.$$ We focus on the large-$\beta$/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in $h$ from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with $\phi(x)=0$. We prove that the critical value for $h$ corresponds to that of the annealed model and is given by $h_c(\alpha)= -\log \mathbb E[e^{\alpha \omega}]$, and that near the critical point, the free energy displays the following critical behavior $$F_\beta(\alpha,h_c+u )\stackrel{u\to 0+}{\sim} \max_{n\ge 1} \left\{\theta_1 e^{-4\beta n} u- \frac{1}{2}\theta^2_1 e^{-8\beta n} \frac{\mathrm{Var}\left[e^{\alpha \omega}\right]}{\mathbb E \left[ e^{\alpha \omega} \right]^2}\right\}.$$ The positive constant $\theta_1(\beta)>0$ is defined by the asymptotic probability of spikes for the infinite volume SOS with $0$ boundary condition $\theta_1(\beta):=\lim_{n\to \infty} e^{4\beta n}\mathbf P_{\beta} (\phi({\bf 0})=n)$ ...