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

TL;DR

This paper studies a random walk model for SOS level lines, establishing a scaling limit and tail bounds that enhance understanding of entropic repulsion phenomena in 3D interfaces.

Contribution

It introduces a new line ensemble model with geometric tilts, proves a 1:2:3 scaling limit, and derives Tracy-Widom tail bounds, addressing a question from Caputo, Ioffe, and Wachtel (2019).

Findings

Established a 1:2:3 scaling limit for the line ensemble.

Derived Tracy-Widom-type upper tail bounds for the curves.

Proved a novel ballot theorem for random walk bridges with broad boundary conditions.

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.