Uniform convergence of Dyson Ferrari--Spohn diffusions to the Airy line ensemble

TL;DR

This paper proves that the entire Dyson Ferrari--Spohn diffusion ensemble converges uniformly to the Airy line ensemble under specific scaling, extending previous results that only considered the top curve.

Contribution

We establish uniform convergence of the full Dyson Ferrari--Spohn diffusion ensemble to the Airy line ensemble, generalizing prior finite-dimensional distribution convergence results.

Findings

Full ensemble convergence to Airy line ensemble

Formulation of a Brownian Gibbs property with area tilts

Equivalence to the usual Brownian Gibbs property after a shift

Abstract

We consider the Dyson Ferrari--Spohn diffusion $\mathcal{X}^N = (\mathcal{X}^N_1,\dots,\mathcal{X}^N_N)$, consisting of $N$ non-intersecting Ferrari--Spohn diffusions $\mathcal{X}^N_1 > \cdots > \mathcal{X}^N_N > 0$ on $\mathbb{R}$. This object was introduced by Ioffe, Velenik, and Wachtel (2018) as a scaling limit for line ensembles of $N$ non-intersecting random walks above a hard wall with area tilts, which model certain three-dimensional interfaces in statistical physics. It was shown by Ferrari and Shlosman (2023) that as $N\to\infty$, after a spatial shift of order $N^{2/3}$ and constant rescaling in time, the top curve $\mathcal{X}^N_1$ converges to the $\mathrm{Airy}_2$ process in the sense of finite-dimensional distributions. We extend this result by showing that the full ensemble $\mathcal{X}^N$ converges with the same shift and time scaling to the Airy line ensemble in the topology of uniform convergence on compact sets. In our argument we formulate a Brownian Gibbs property with area tilts for $\mathcal{X}^N$, which we show is equivalent after a global parabolic shift to the usual Brownian Gibbs property introduced by Corwin and Hammond (2014).