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 λ pointing down (the prewetting problem). In 1996, Cesi and Martinelli showed that if the inverse-temperature β is large enough, then along a decreasing sequence of critical points (λc(k))_k=0K^β the dynamics is torpid: its inverse spectral gap is O(1) when λ∈(λ_c^(k+1),λ_c^(k)) whereas it is exp[Θ(n)] at each λ_c^(k) for each k≤K_β, due to a coexistence of rigid phases at heights k+1 and k. Our focus is understanding (a) the onset of metastability as λ_n↑λ_c^(k); and (b) the effect of an unbounded number of layers, as we remove the restriction k≤K_β, and even allow for λ_n→0 towards the λ=0 case which has O(logn) 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[Θ~(1/|λ_n-λ_c^(k)|)] as λ↑λ_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 λ_n=n^-α (corresponding to k_n≍logn), this also interpolates down to the behavior of the dynamics when λ=0. We complement this by extending the fast mixing to all λ uniformly bounded away from (λc(k))_k=0^∞. Together, these results provide a sharp understanding of the predicted infinite sequence of dynamical phase transitions governed by the layering phenomenon.