On the limiting law of line ensembles of Brownian polymers with geometric area tilts

TL;DR

This paper proves the convergence of line ensembles of non-crossing Brownian bridges with geometric area tilts to a limiting law, addressing an open problem and extending previous results on their behavior under free boundary conditions.

Contribution

It establishes the limit law of the top paths of Brownian line ensembles with geometric area tilts under free boundary conditions, confirming a conjecture from prior work.

Findings

Top $k$ paths converge to a limit as interval length and number of paths grow.

The limit matches the free boundary case conjectured earlier.

Addresses an open problem in the theory of Brownian line ensembles.

Abstract

We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of the law of the top $k$ paths, for any fixed $k$, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to $\infty$, the top $k$ paths converge to the same limit as in the free boundary case, as conjectured by Caputo, Ioffe and Wachtel.