Uniqueness, mixing, and optimal tails for Brownian line ensembles with geometric area tilt

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 (2+1)-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.