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

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 existence of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to ∞, the top k paths converge to the same limit as in the zero boundary case, as conjectured by Caputo, Ioffe and Wachtel.