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