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

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 μ. 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 μ as $N\to \mathrm{\infty }$ followed by $n\to \mathrm{\infty }$. We do so both in the case of bridges fixed at $\pm N$ and of walks fixed only at $-N$.