Asymptotic Phenomena in the Six-Vertex Model

Abstract

In this thesis we establish several results concerning the large scale geometry of six-vertex models. These include limit shape phenomena, fluctuation theorems, convergence of local statistics, and boundary-induced phase transitions. This work is divided into three parts. The first part analyzes height fluctuations for the stochastic six-vertex model under a translation-invariant boundary condition. On a grid of diameter T, we show these fluctuations are of order T^(1/3) along a certain characteristic line and converge to the Baik-Rains distribution after scaling; the latter is widely believed to govern fluctuations for stationary Markov processes in the Kardar-Parisi-Zhang universality class. An archetypal model in this class is the asymmetric simple exclusion process (ASEP), which is a limiting case of the stochastic six-vertex model. So, our methods further enable us to show analogous current fluctuation theorems for the stationary ASEP. The second part analyzes limit shapes and local statistics for the stochastic six-vertex model on a cylinder with arbitrary initial data. We show that it exhibits a limit shape, whose density profile is given by the entropy solution to an explicit, non-linear conservation law that was predicted by Gwa-Spohn and by Reshetikhin-Sridhar. We moreover show that the local statistics of this model around any continuity point of its limit shape converge to one of the translation-invariant six-vertex models considered in the first part of the thesis. The third part concerns boundary-induced phase transitions (also called arctic boundaries) for the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of a uniformly random alternating sign matrix. We show that it exhibits the arctic boundary phenomenon, with the boundary given by a union of explicit algebraic curves. This was originally predicted by Colomo-Sportiello as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case.