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 $\exp[\Theta(n)]$ at each $\lambda_c^{(k)}$ for each $k\leq K_\beta$, 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 $\exp[\tilde\Theta(1/|\lambda_n-\lambda_c^{(k)}|)]$ 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 compliment this by extending the fast mixing to all $\lambda$ uniformly bounded away from $(\lambda_c^{(k)})_{k=0}^\infty$. Together, these results provide a sharp understanding of the predicted infinite sequence of dynamical phase transitions governed by the layering phenomenon.