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

TL;DR

This paper studies a non-integrable model of non-colliding Brownian lines with geometric area tilts, establishing its ergodic properties, tail estimates, and uniqueness of the infinite volume Gibbs measure, advancing understanding of KPZ universality class interfaces.

Contribution

It proves ergodicity, decay of correlations, tail bounds, and uniqueness of the Gibbs measure for the $ ext{lambda}$-tilted line ensemble, resolving open questions in the field.

Findings

Proved the zero boundary LE is mixing and ergodic.

Established optimal upper tail estimates matching the Ferrari-Spohn diffusion.

Proved the uniqueness of the Gibbs measure among uniformly tight line ensembles.

Abstract

We consider non-colliding Brownian lines above a hard wall, which are subject to geometrically growing (given by a parameter $\lambda>1$) area tilts, which we call the $\lambda$-tilted line ensemble (LE). The model was introduced by Caputo, Ioffe, Wachtel [CIW] in 2019 as a putative scaling limit for the level lines of low-temperature 3D Ising interfaces. While the LE has infinitely many lines, the case of the single line, known as the Ferrari-Spohn (FS) diffusion, is one of the canonical interfaces appearing in the Kardar-Parisi-Zhang (KPZ) universality class. In contrast with well studied models with determinantal structure such as the Airy LE constructed by Corwin and Hammond as well as the FS diffusion, the $\lambda$-tilted LE is non-integrable. [CIW] constructed a stationary infinite volume Gibbs measure (the zero boundary LE) as a limit of finite LEs on finite intervals with zero boundary conditions, and obtained control on its fluctuations in terms of first moment estimates. Subsequently, Dembo, Lubetzky, Zeitouni revisited the case of finitely many lines and established an equivalence between the free and the zero boundary LEs. In this article we develop probabilistic arguments to resolve several questions that remained open. We prove that the infinite volume zero boundary LE is mixing and hence ergodic and establish a quantitative decay of correlation. Further, we prove an optimal upper tail estimate for the top line matching that of the FS diffusion. Finally, we prove uniqueness of the Gibbs measure in the sense that any uniformly tight LE (a notion which includes all stationary $\lambda$-tilted LE) must be the zero boundary LE. This immediately implies that the LE with free boundary conditions, as the number of lines and the domain size go to infinity arbitrarily converges to this unique LE.