On level line fluctuations of SOS surfaces above a wall

We study the low-temperature (2 + 1)D solid-on-solid model on 1, L² with zero boundary conditions and nonnegative heights (a floor at height 0). Caputo et al. (2016) established that this random surface typically admits either or + 1 many nested macroscopic level line loops {L}≥₀ for an explicit log L, and its top loop L₀ has cube-root fluctuations: For example, if () is the vertical displacement of L₀ from the bottom boundary point (, 0), then max () = L¹/3+(1⁾ over ∈ I₀ := L/2+−L²/³, L²/³. It is believed that rescaling by L¹/³ and I₀ by L²/³ would yield a limit law of a diffusion on [−1, 1]. However, no nontrivial lower bound was known on () for a fixed ∈ I₀ (e.g., = ₂ ), let alone on min () in I₀, to complement the bound on max (). Here, we show a lower bound of the predicted order L¹/³: For every > 0, there exists > 0 such that min∈₀ () ≥ L¹/³ with probability at least 1−. 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 L²/³ × L²/³ box with boundary conditions − 1, , , (i.e., − 1 on the bottom side and elsewhere), the limit of () as , L → ∞ is a Ferrari–Spohn diffusion.