Exponentially slow Mixing arising from Entropic Repulsion in $p$-SOS model

We investigate the Glauber dynamics of the generalized (2+1)-dimensional $p$-SOS model, $1<p<\infty$, under a hard floor constraint. This constraint induces entropic repulsion: the integer-valued interface is forced above the wall and rises to a typical height $H(p,L)$ depending on both $p$ and side length $L$. Entropic repulsion has been studied in many random interface models. For the classical SOS model ($p=1$), previous works proved an exponential lower bound on the mixing time in the low-temperature regime (large inverse temperature $\beta$). Beyond this case, however, no rigorous dynamical lower bounds were known. Even for the Discrete Gaussian model ($p=2$), the metastable slowdown predicted by the entropic-repulsion picture remained open. On the equilibrium side, sharp large-deviation principles and precise estimates of typical and maximal heights were known for all $1\le p<\infty$, but their dynamical consequences had not been established.
Our main contribution is to close this gap by proving that slow mixing arising from entropic repulsion persists throughout $1<p<\infty$. Specifically, we establish the stretched-exponential lower bound $\tau_{\mathrm{mix}}\ge \exp(cL^{1-o(1)})$, for some $c>0$ depending on $p$ and $\beta$. We also give a refined metastability analysis: for every $0<a<1$, the hitting time of the intermediate level $aH(p,L)$ is at least $\exp(cL^{a^{d(p)}-o(1)})$, where $d(p)>0$ depends on $p$. The proof extends the Peierls-type contour estimates developed for $p=1$ to the nonlinear $p$-SOS setting. These results show that entropic repulsion induces uniformly slow mixing across the full $p$-SOS family, extending a phenomenon previously established only for $p=1$.