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 \(k\le K_\beta \) , 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 \(\lambda _n\rightarrow 0\) towards the \(\lambda = 0\) case which has \(O(\log n)\) layers and was studied by Caputo et al. (Ann Probab 42(4):1516-1589, 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 complement 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.
