Scaling limit and tail bounds for a random walk model of SOS level lines

Abstract

This paper analyzes a random walk model for the level lines appearing in the entropic repulsion phenomena of three-dimensional discrete random interfaces above a hard wall; we are particularly motivated by the low-temperature (2+1)D solid-on-solid (SOS) model, where the emergence of these level lines has been rigorously established. The model we consider is a line ensemble of non-crossing random walk bridges above a wall with geometrically growing area tilts. Our main result, which in particular resolves a question of Caputo, Ioffe, and Wachtel (2019), is an edge 1:2:3 scaling limit for this ensemble as the domain size $N$ diverges, with a growing number of walks (including the number of level lines of the SOS model) and high boundary conditions (covering the maximum upper deviation of the SOS level lines). As a key input, we establish Tracy--Widom-type upper tail bounds for each of the relevant curves in the line ensemble. An ingredient which may be of independent interest is a ballot theorem for random walk bridges under a broader range of boundary values than available in the literature.

Figures (7)

• Figure 1: Left: an illustration of the limit shape of the SOS level lines after rescaling to $[-1,1]^2$. The loops $\frac{1}{N}\gamma_i$ converge to the nested Wulff shapes shown here. The limit shape is flat for all level lines on central $(1-\delta_\beta)$-portions of each side of the box (cyan), where $\delta_\beta\downarrow 0$ as $\beta\uparrow\infty$. Right: the effective random walk model \ref{['areatiltRW']} of the level line fluctuations about their flat limit shape, as seen by zooming in on the blue dashed region in the figure on the left. In this region the level lines appear as a stack of ordered open contours above a floor. In an effective approximation one assumes these contours to be height functions with respect to the horizontal axis (ignoring possible microscopic overhangs), as shown here.
• Figure 2: To lower bound $\mathbb{P}^{AH_\lambda, AH_\lambda}_{I;0}(X(J-J_A)\leq H_\lambda)$, we force the path to fall linearly as depicted above. Note that across the interval of size $J_A = A^{1/2}H_\lambda^2$, the path then falls by an amount $AH_\lambda$, which is not diffusive with respect to the interval length. However, the path does fall by an on-scale amount on the smaller scale subintervals of length $A^{-1/2}H_\lambda$, which allows us to make use of invariance principles. We expect that in the true behavior, the path would fall according to a parabola with curvature $\lambda$, but we do not care about getting the sharp coefficient in the exponent in the lower bound for the overall partition function $Z_{I;0}^{\lambda;AH_\lambda,BH_\lambda}$, and the above prescription is simpler to analyze and achieves the correct order.
• Figure 3: For $j\leq m$, $\mathsf{Cl}_{j+1}$ (red) features a very high portion outside $\mathcal{I}_{j+1}$, coming from a global max bound, and a portion that grows like ${\varepsilon_j^{-2}H_{j+1}(\log\frac{|x|}{K^{1/2}H_{j+1}^2})^{2/3}}$ inside $\mathcal{I}_{j+1}$. The random walk bridge (orange) with area tilt $\lambda_j$, boundary conditions $u_j, v_j \leq BN^{1/3}$, and floor at $\mathsf{Cl}_{j+1}$ is shown in Proposition \ref{['prop:recursive-bound']} to stay below $\mathsf{Cl}_j$ (blue) with high probability.
• Figure 4: An illustration of the application of \ref{['lem:drop-high-bc']} to Proposition \ref{['prop:recursive-bound']} and the proof of \ref{['lem:drop-high-bc']}. The floor at $\mathsf{Cl}_{j+1}$ features a potentially large gap at $x_{j+1}^{\mathrm{L}}$ and $x_{j+1}^{\mathrm{R}}$. The first step is to show that, within $[x_{j+1}^{\mathrm{L}},x_{j}^{\mathrm{L}}]$ and $[x_j^{\mathrm{R}} , x_{j+1}^{\mathrm{R}}]$, the random walk drops to $\approx \varepsilon_j^{-1}H_j$ above $\mathsf{Cl}_{j+1}(x_{j+1}^{\mathrm{R}}) = \mathsf{Cl}_{j+1}(x_{j+1}^{\mathrm{L}}) = \max_{x \in \mathcal{I}_{j+1}} \mathsf{Cl}_{j+1}(x)$ (blue). The max bound (Lemma \ref{['lem:rec-global-max-bound']}) is used to show $X(x_{j+1}^{\mathrm{L}})\vee X(x_{j+1}^{\mathrm{R}})\leq BN^{1/3}$, see \ref{['eqn:max of X for figure']}. Lemma \ref{['lem:drop-high-bc']} then gives random times $x^{\mathrm{L}}, x^{\mathrm{R}} \in \mathcal{I}_{j+1}\setminus \mathcal{I}_j$ where $X$ lies roughly $\varepsilon_j^{-1}H_j$ above the floor (the two innermost orange points in the figure). For the proof of \ref{['lem:drop-high-bc']}, note the boundary conditions at $BN^{1/3}$ may be as large as $N^{1-\varepsilon}$ for any $\varepsilon>0$, much higher than the blue line at $\mathsf{Cl}_{j+1}(x_{j+1}^R)$; in particular, the dropping lemma estimate \ref{['eqn:dropping-lemma']} is not sufficient to immediately produce $x^{\mathrm{L}}$ and $x^{\mathrm{R}}$. Thus, we apply the dropping lemma in an iterative manner to the random walk on $\mathcal{I}_{j+1}$ with floor at $\mathsf{Cl}_{j+1}(x^{\mathrm{L}}_{j+1})$, yielding a sequence of random drop points at decreasing heights $(A_i H_j)_{i\geq 1}$ (orange, see \ref{['claim:drop-high-bc-input']}) until $x^{\mathrm{L}}$ and $x^{\mathrm{R}}$ are produced.
• Figure 5: Step 1 involves bounding the walk $X$ at a sequence of mesh points $x(0) > x(1) > \cdots$ and $-x(0) < -x(1) < \cdots$ by a quantity $\mathfrak{B}_j(k)$ at $\pm x(k)$. Suppose we have already shown $X(-x(k)) \vee X(x(k)) \leq \mathfrak{B}_j(k)$, for some $k$ (outermost red dots). Restrict the walk to $I(k) = [-x(k), x(k)]$ and raise the floor from $\mathsf{Cl}_{j+1}$ (black curve) to $\mathsf{Cl}_{j+1}(x(k))$ (blue). Use the dropping lemma to produce points $z_1^{\mathrm{R}}(k+1)$ and $z_2^{\mathrm{R}}(k+1)$ to the left and right of $x(k+1)$, where the walk has dropped to $\leq D\varepsilon_j^{-1}H_j$ above the blue floor (orange dots). Restrict the walk to $[z_2^{\mathrm{R}}(k+1), z_1^{\mathrm{R}}(k+1)]$ (orange, dashed), raise the boundary conditions to $D\varepsilon_j^{-1}H_j + \mathsf{Cl}_{j+1}(x(k))$, and use the one-point bound (\ref{['oneptbd']}) to show $X(x(k+1))\leq \mathfrak{B}_j(k+1)$ with high probability. Step 2 of the proof involves restricting the walk to $[x(k+1), x(k)]$ (red, dashed), raising the boundary conditions to $\mathfrak{B}_j(k)$ (which exceeds $\mathfrak{B}_j(k+1)$, red dots), and then using the max bound \ref{['maxbd']} to show that, w.h.p., $\max_{x \in [x(k+1), x(k)]} X(x)$ is bounded by a quantity which is less than $\mathsf{Cl}_j(x(k+1))= \min_{x \in [x(k+1), x(k)]}\mathsf{Cl}_j(x)$. By symmetry, the same argument applies on the left side of $0$.

Theorems & Definitions (89)

• Theorem 1
• Theorem 2
• Remark 1.1
• Remark 1.2: Parabolic decay of the random walks
• Remark 1.3: Range of parameters, relevance to SOS level lines
• Remark 2.3
• Definition 2.4
• Remark 2.5: Notation for measures
• Remark 2.6
• Remark 2.7