Characterizing Gibbs states for area-tilted Brownian lines

Gibbsian line ensembles are families of Brownian lines arising in many natural contexts such as the level curves of three dimensional Ising interfaces, the solid-on-solid model, multilayered polynuclear growth, trajectories of eigenvalues as the entries of the corresponding matrix perform diffusions, to name a few. In particular, line ensembles with area tilt potentials play a significant role in the study of wetting and entropic repulsion phenomena. An important example is a class of nonintersecting Brownian lines above a hard wall, which are subject to geometrically growing area tilt potentials, which we call the lambda-tilted line ensemble, where lambda > 1 is the parameter governing the geometric growth. The model was introduced in (Electron. J. Probab. 24 (2019) 37) as a putative scaling limit of the level lines of entropically repulsed solid-on-solid interfaces. While this model has infinitely many lines and is nonintegrable, the case of the single line, known as the Ferrari-Spohn diffusion, is well studied (Ann. Probab. 33 (2005) 1302-1325). In this article we address the problem of classifying all Gibbs measures for )-tilted line ensembles. A stationary infinite volume Gibbs measure was already constructed in (Electron. J. Probab. 24 (2019) 37; In Statistical Mechanics of Classical and Disordered Systems (2019) 241-266 Springer), and the uniqueness of this translation invariant Gibbs measure was established in (Probab. Math. Phys. 6 (2025) 195-239). Our main result here is a strong characterization for Gibbs measures of )-tilted line ensembles in terms of a two parameter family. Namely, we show that the extremal Gibbs measures are completely characterized by the behavior of the top line X1 at positive and negative infinity, which must satisfy the parabolic growthX-1(t) = t(2) +L vertical bar t vertical bar 1(t<0) +R vertical bar t vertical bar 1(t>0) +o(vertical bar t vertical bar) as vertical bar t vertical bar -> infinity,where L, R are real parameters, including-infinity, with L + R <0, while the lower lying lines remain uniformly confined. The case L = R = -infinity corresponds to the unique translation invariant Gibbs measure. The result bears some analogy to the Airy wanderers, an integrable model introduced and studied in (Ann. Probab. 38 (2010) 714-769) in the context of the Airy line ensemble, except in this case only the top line can wander, owing to the geometrically increasing nature of the area tilting factor. A crucial step in our proof, which holds independent significance, is a complete characterization of the extremal Gibbs states associated to a single area-tilted Brownian excursion, describing the equilibrium states of the top line when the lower lines are absent, and which can be interpreted as nontranslation invariant versions of the Ferrari-Spohn diffusion.