The stationary horizon as the central multi-type invariant measure in the KPZ universality class

The Kardar-Parisi-Zhang (KPZ) universality class describes a large class of 2-dimensional models of random growth, which exhibit universal scaling exponents and limiting statistics. The last ten years has seen remarkable progress in this area, with the formal construction of two interrelated limiting objects, now termed the KPZ fixed point and the directed landscape (DL). This dissertation focuses on a third central object, termed the stationary horizon (SH). The SH was first introduced (and named) by Busani as the scaling limit of the Busemann process in exponential last-passage percolation. Shortly after, in the author's joint work with Sepp\"al\"ainen, it was independently constructed in the context of Brownian last-passage percolation. In this dissertation, we give an alternate construction of the SH, directly from the description of its finite-dimensional distributions and without reference to Busemann functions. From this description, we give several exact distributional formulas for the SH. Next, we show the significance of the SH as a key object in the KPZ universality class by showing that the SH is the unique coupled invariant distribution for the DL. A major consequence of this result is that the SH describes the Busemann process for the DL. From this connection, we give a detailed description of the collection of semi-infinite geodesics in the DL, from all initial points and in all directions. As a further evidence of the universality of the SH, we show that it appears as the scaling limit of the multi-species invariant measures for the totally asymmetric simple exclusion process (TASEP). This dissertation is adapted from two joint works with Sepp\"al\"ainen and two joint works with Busani and Sepp\"al\"ainen.Comment: v2: minor typos corrected, PhD dissertation, University of Wisconsin--Madison (2023). Contains material adapted from arXiv:2103.01172, arXiv:2112.10729, arXiv:2203.13242 and arXiv:2211.04651. Chapter 3 gives an alternate proof of the invariance of the SH shown in arXiv:2203.1324