Metastability cascades and prewetting in the SOS model

We study Glauber dynamics for the low temperature $(2+1)$D Solid-On-Solid model on a box of side-length $n$ with a floor at height $0$ (inducing entropic repulsion) and a competing bulk external field $\lambda$ pointing down (the prewetting problem). In 1996, Cesi and Martinelli showed that if the inverse-temperature $\beta$ is large enough, then along a decreasing sequence of critical points $({\lambda }_c^{(k)}{)}_{k=0}^{K_{\beta }}$ the dynamics is torpid: its inverse spectral gap is $O(1)$ when $\lambda \in ({\lambda }_c^{(k+1)},{\lambda }_c^{(k)})$ whereas it is $\exp [\mathrm{\Theta }(n)]$ at each ${\lambda }_c^{(k)}$ for each $k\le K_{\beta }$, due to a coexistence of rigid phases at heights $k+1$ and $k$. Our focus is understanding (a) the onset of metastability as ${\lambda }_n\uparrow {\lambda }_c^{(k)}$; and (b) the effect of an unbounded number of layers, as we remove the restriction $k\le K_{\beta }$, and even allow for ${\lambda }_n\to 0$ towards the $\lambda =0$ case which has $O(\log n)$ layers and was studied by Caputo et al. (2014). We show that for any $k$, possibly growing with $n$, the inverse gap is $\exp [\tilde{\mathrm{\Theta }}(1\mathrm{/}\mathrm{\mid }{\lambda }_n-{\lambda }_c^{(k)}\mathrm{\mid })]$ as $\lambda \uparrow {\lambda }_c^{(k)}$ up to distance $n^{-1+o(1)}$ from this critical point, due to a metastable layer at height $k$ on the way to forming the desired layer at height $k+1$. By taking ${\lambda }_n=n^{-\alpha }$ (corresponding to $k_n\asymp \log n$), this also interpolates down to the behavior of the dynamics when $\lambda =0$. We complement this by extending the fast mixing to all $\lambda$ uniformly bounded away from $({\lambda }_c^{(k)}{)}_{k=0}^{\mathrm{\infty }}$. Together, these results provide a sharp understanding of the predicted infinite sequence of dynamical phase transitions governed by the layering phenomenon.