Characterizing Gibbs states for area-tilted Brownian lines

TL;DR

This paper classifies all Gibbs measures for a class of non-intersecting Brownian line ensembles with area tilt potentials, revealing a two-parameter family characterized by the asymptotic behavior of the top line.

Contribution

It provides a complete characterization of extremal Gibbs measures for $\lambda$-tilted line ensembles, including the translation-invariant case and non-invariant cases with specific asymptotics.

Findings

Extremal Gibbs measures are characterized by parameters L and R governing the top line's growth.

The unique translation-invariant Gibbs measure corresponds to L=R=- finite.

The results relate to models like Airy wanderers and Ferrari-Spohn diffusion.

Abstract

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, multi-layered polynuclear growth etc. An important example is a class of non-intersecting Brownian lines above a hard wall, which are subject to geometrically growing area tilt potentials, which we call the $\lambda$-tilted line ensemble, where $\lambda>1$. The model was introduced by Caputo, Ioffe and Wachtel [CIW] in 2018, as a putative scaling limit of the level lines of entropically repulsed solid-on-solid interfaces. In this article we address the problem of classifying all Gibbs measures for $\lambda$-tilted line ensembles. A stationary infinite volume Gibbs measure was already constructed by [CIW], and the uniqueness of this translation invariant Gibbs measure was recently established by Caputo and Ganguly. Our main result here is a strong characterization for Gibbs measures of $\lambda$-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 $X^1$ at positive and negative infinity, which must satisfy the parabolic growth $$X^{1}(t)=t^2+L\,|t|\,\mathbf{1}_{t<0}+R\,|t|\,\mathbf{1}_{t>0}+o(|t|)\,,\quad \text{ as } \;|t| \to \infty,$$ where $L,R$ are real parameters, including $-\infty,$ with $L+R<0$. The case $L=R=-\infty$ corresponds to the unique translation invariant Gibbs measure. The result bears some analogy to the Airy wanderers, an integrable model introduced and studied in the context of the Airy line ensemble. A crucial step in our proof, of independent significance, is a complete characterization of the extremal Gibbs states associated to a single area-tilted Brownian excursion, which can be interpreted as non-translation invariant versions of the Ferrari-Spohn diffusion.