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

TL;DR

The paper develops a probabilistic framework for λ-tilted line ensembles with geometric area tilts (λ>1) and studies their infinite-volume limits under zero and free boundary conditions. It proves that the infinite-volume zero-boundary LE μ^0 is mixing and ergodic with a quantitative decay of correlations, establishes an essentially optimal upper tail for the top line matching Ferrari-Spohn (FS) diffusion, and proves uniqueness: any uniformly tight, asymptotically pinned λ-tilted LE must be μ^0, implying convergence of free-boundary LEs to μ^0. The approach avoids integrable structure, relying on Brownian-Gibbs resampling and coupling techniques, including a reverse coupling between free and zero boundary ensembles and a bootstrap from uniform confinement to stretched-tail estimates. These results illuminate the universal behavior of non-integrable GEOMETRIC-tilted line ensembles and provide a robust framework for studying tail behavior, ergodicity, and uniqueness in infinite-volume Gibbs-type line ensembles.

Abstract

We consider non-colliding Brownian lines above a hard wall, which are subject to geometrically growing (given by a parameter $λ>1$) area tilts, which we call the $λ$-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 $λ$-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 $λ$-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.

Figures (11)

• Figure 3.1: An illustration of the resampling argument outlined. On the left, we have two independent samples of $\mu^{0}_{n,T}$ (pink) and $\mu^{f}_{n,T}$ (green) with $n=4$. The stopping domain where the pink data dominates the green data is denoted by $[\tau_{\ell}, \tau_r]$. On the right, we have the resampled ensembles (denoted by orange and blue respectively) under the monotone coupling causing the orange curves to deterministically be above the blue curves.
• Figure 4.1: The ensemble pinned at zero at gaps of $2$ with finitely many lines (four in the figure) owing to random fluctuations will be above the infinite ensemble with the same number of lines at the random locations $\tau_{\ell}, \tau_r,$ as in the proof.
• Figure 5.1: The trapezoids $\mathcal{T}_i$ and $\mathcal{T}_{i+1}$.
• Figure 5.2: A sketch of the event in \ref{['eq:stretchedexp8']} for $i=K$. Here $S:=T-\ell$, $h=\log^5(y)$.
• Figure 6.1: Illustration of the proof strategy for Theorem \ref{['th:mixing']}. The ensemble $\underline{Y}$ pinned at $\pm t/2$ and ${3t}/{2}$ (green lines) is monotonically coupled to $\underline{X}$ with law $\mu^{0}$ (pink lines). The weak convergence guarantees that the gap between the top green lines and the top pink lines goes to zero in expectation on compact domains centered around $0$ and $t.$ Further, the green lines on $[-t/2,t/2]$ and $[t/2,3t/2]$ are independent due to the pinning.

Theorems & Definitions (50)

• Definition 2.1: $\lambda$-tilted LE
• Definition 2.2: Asymptotically pinned to zero
• Lemma 2.3
• Remark 2.4
• Lemma 2.5
• Remark 2.6
• Theorem 2.7
• Corollary 2.8
• Remark 2.9
• Theorem 2.10