  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.
