Characterization of $H$-Brownian Gibbsian line ensembles

TL;DR

This paper proves that $H$-Brownian Gibbsian line ensembles are uniquely determined by the finite-dimensional distributions of their lowest curve, with a key application to the KPZ line ensemble and its characterization.

Contribution

It establishes a complete characterization of $H$-Brownian Gibbsian line ensembles via the finite-dimensional marginals of their lowest curve, including the uniqueness of the KPZ line ensemble.

Findings

$H$-Brownian Gibbsian line ensembles are characterized by the lowest curve's marginals.

The KPZ line ensemble is uniquely identified by its Gibbs property and lowest curve.

The result applies to a broad class of interaction Hamiltonians $H$.

Abstract

In this paper we show that an $H$-Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional marginals of its lowest indexed curve for a large class of interaction Hamiltonians $H$. A particular consequence of our result is that the KPZ line ensemble is the unique line ensemble that satisfies the $H_1$-Brownian Gibbs property with $H_1(x) = e^x$ and whose lowest indexed curve is equal to the Hopf-Cole solution to the narrow wedge KPZ equation.