We consider noncolliding Brownian lines above a hard wall, subject to
geometrically growing self-potentials of tilted area type. The model was
proposed by Caputo, Ioffe, and Wachtel as the scaling limit for the level lines of
-dimensional
solid-on-solid random interfaces above a hard wall. In contrast with the well-studied
Airy line ensemble, a central object in the KPZ universality class, the presence of
growing area tilts renders the model nonintegrable. A stationary infinite-volume
Gibbs measure was previously constructed as a limit of finite line ensembles on finite
intervals with zero boundary conditions. We refer to this as the zero boundary state.
Some preliminary control on its fluctuations was given in terms of first moment
estimates for one-point marginals and for suitable curved maxima. Subsequently,
Dembo, Lubetzky, and Zeitouni revisited the case of finitely many lines and
established an equivalence between the free and the zero boundary states. We develop
probabilistic arguments to resolve several questions that remained open. We prove
that the zero boundary state is mixing, and hence ergodic, and establish
a quantitative decay of correlation. Further, we prove an optimal upper
tail estimate for the top line showing that it behaves approximately as a
Ferrari–Spohn diffusion, which corresponds to the process obtained by neglecting all
interactions with lower lying lines. Finally, we prove that there exists a unique
uniformly tight Gibbs measure, which implies uniqueness of the stationary
state, and convergence to this state of the free boundary ensembles as the
number of lines and the domain size are taken to infinity in an arbitrary
fashion.
