On level line fluctuations of SOS surfaces above a wall

Abstract

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 $\{{\mathcal{L}}_i{\}}_{i\ge 0}$ 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., $x=\frac{L}{2}$), 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 ${\min }_{x\in I_0}\rho (x)\ge \delta L^{1/3}$ 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.

arXiv comment63 pages, 2 figures. Final version

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Cited by in corpus (1)

• Uniqueness, mixing, and optimal tails for Brownian line ensembles with geometric area tilt