# Scaling limit for line ensembles of random walks with geometric area tilts

## TL;DR

This paper proves the existence of a scaling limit for non-intersecting random walk line ensembles with geometric area tilts, connecting discrete models to continuous limiting measures as the interval and number of paths grow large.

## Contribution

It establishes the convergence of the top paths of random walk ensembles with geometric area tilts to a universal limit measure under mild assumptions.

## Key findings

- Top paths converge to a limiting measure as interval and path number grow.
- Results hold for both fixed-end bridges and walks fixed only at one end.
- Provides rigorous proof for the open problem of scaling limits in this model.

## Abstract

We consider line ensembles of non-intersecting random walks constrained by a hard wall, each tilted by the area underneath it with geometrically growing pre-factors $\mathfrak{b}^i$ where $\mathfrak{b}>1$. This is a model for the level lines of the $(2+1)$D SOS model above a hard wall, which itself mimics the low-temperature 3D Ising interface. A similar model with $\mathfrak{b}=1$ and a fixed number of curves was studied by Ioffe, Velenik, and Wachtel (2018), who derived a scaling limit as the time interval $[-N,N]$ tends to infinity. Line ensembles of Brownian bridges with geometric area tilts ($\mathfrak{b}>1$) were studied by Caputo, Ioffe, and Wachtel (2019), and later by Dembo, Lubetzky, and Zeitouni (2022). Their results show that as the time interval and the number of curves $n$ tend to infinity, the top $k$ paths converge to a limiting measure $\mu$. In this paper we address the open problem of proving existence of a scaling limit for random walk ensembles with geometric area tilts. We prove that with mild assumptions on the jump distribution, under suitable scaling the top $k$ paths converge to the same measure $\mu$ as $N\to\infty$ followed by $n\to\infty$. We do so both in the case of bridges fixed at $\pm N$ and of walks fixed only at $-N$.