On level line fluctuations of SOS surfaces above a wall

We study the low temperature \((2+1)\)D Solid-On-Solid model on \([[1, L ]]^2\) with zero boundary conditions and nonnegative heights (a floor at height \(0\)). Caputo et al. (2016) established that this random surface typically admits either \(\mathfrak h \) or \(\mathfrak h+1\) many nested macroscopic level line loops \(\{\mathcal L_i\}_{i\geq 0}\) for an explicit \(\mathfrak h\asymp \log L\), and its top loop \(\mathcal L_0\) has cube-root fluctuations: e.g., if \(\rho(x)\) is the vertical displacement of \(\mathcal L_0\) from the bottom boundary point \((x,0)\), then \(\max \rho(x) = L^{1/3+o(1)}\) over \(x\in I_0:=L/2+[[-L^{2/3},L^{2/3}]]\). It is believed that rescaling \(\rho\) by \(L^{1/3}\) and \(I_0\) by \(L^{2/3}\) would yield a limit law of a diffusion on \([-1,1]\). However, no nontrivial lower bound was known on \(\rho(x)\) for a fixed \(x\in I_0\) (e.g., \(x=\frac L2\)), let alone on \(\min\rho(x)\) in \(I_0\), to complement the bound on \(\max\rho(x)\). Here we show a lower bound of the predicted order \(L^{1/3}\): for every \(\epsilon>0\) there exists \(\delta>0\) such that \(\min_{x\in I_0} \rho(x) \geq \delta L^{1/3}\) with probability at least \(1-\epsilon\). The proof relies on the Ornstein--Zernike machinery due to Campanino-Ioffe-Velenik, and a result of Ioffe, Shlosman and Toninelli (2015) that rules out pinning in Ising polymers with modified interactions along the boundary. En route, we refine the latter result into a Brownian excursion limit law, which may be of independent interest. We further show that in a \( K L^{2/3}\times K L^{2/3}\) box with boundary conditions \(\mathfrak h-1,\mathfrak h,\mathfrak h,\mathfrak h\) (i.e., \(\mathfrak h-1\) on the bottom side and \(\mathfrak h\) elsewhere), the limit of \(\rho(x)\) as \(K,L\to\infty\) is a Ferrari--Spohn diffusion.