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 under a hard floor constraint. This setting induces entropic repulsion: the integer-valued interface height is forced to remain above the wall and consequently rises to a typical height $H(p,L)$ that depends on both the parameter $p$ and the system size $L$. In the classical SOS model ($p=1$), \cite{caputo2016scaling, caputo2014dynamics} derived an exponential lower bound for the mixing time, demonstrating that the Glauber dynamics mixes only after an exponentially long time in the low-temperature regime (large $\beta$, the inverse temperature). However, beyond this case, no rigorous lower bounds were previously known: even for the widely studied Discrete Gaussian model ($p=2$), the metastable slowdown predicted by the entropic repulsion picture had remained an open problem. On the equilibrium side, \cite{lubetzky2016harmonic} obtained sharp large-deviation principles and precise estimates of typical and maximal heights for all $1\le p<\mathrm{\infty }$, but the dynamical consequences of these results had not been established. Our main contribution is to close this gap by proving, for the first time, that exponentially slow mixing arising from entropic repulsion persists throughout the regime $1<p<\mathrm{\infty }$. Specifically, we establish an exponential lower bound, showing that the mixing time satisfies ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{x}}\ge e^{cL}$ for some $c>0$ depending on $p$ and $\beta$. In addition, we provide a refined metastability analysis, proving that the hitting time of an intermediate level $aH(p,L)$ is at least $\exp \left(c{L}^{a^{d(p)}}\right)$, where $d(p)$ is a positive function depending on $p$. Taken together, these results demonstrate that entropic repulsion induces uniformly slow mixing across the entire $p$-SOS family, thereby extending a phenomenon that had previously been established only for $p=1$.