Characterization of H-Brownian Gibbsian line ensembles

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.
Comment: 47 pages, 6 figures. Fixed some typos in the previous version of the paper. arXiv admin note: text overlap with arXiv:2002.00684