Stochastic Dynamics on Integrable Lattice Models

Abstract

The purpose of this thesis is to present some new results related to the six-vertex and dimer model. One theme is the construction and analysis of Markov processes which are naturally associated to these lattice models. Certain integrability properties of the six-vertex and dimer model, often related to the Yang–Baxter equation, allow for the construction of associated Markov chains. In some cases, these are measure preserving Markov chains on configurations of the lattice model. In other cases, they arise via transfer matrices, after choosing a distinguished time coordinate, as a continuous time degeneration of the “time evolution” of the lattice model itself. It is often the case that the integrability of the underlying lattice model provides powerful tools to study the associated Markov chains or their marginals, which are sometimes Markov chains themselves. In Chapter 2, we construct Markov chains on six-vertex states in the quarter plane Z² ≥₀ and the full plane Z². When viewing the six-vertex model as a model of random surfaces, the Markov chain is an example of a surface growth model in the (2+1)-dimensional “Anisotropic KPZ” (or “AKPZ”) universality class. In the Z² case, the translation invariant Gibbs measures of the stochastic sixvertex model are stationary measures of the process. Using structure preserving local moves for the dimer model, in Chapter 3 we construct another surface growth model in the AKPZ universality class, which has the dimer model Gibbs measures as stationary distributions. By exactly computing key quantities such as the current, we confirm predictions from the physics literature on the AKPZ universality class, and we confirm the expected hydrodynamic limit PDE of the growth process in special domains known as tower graphs. To complement our analysis of the growth process, we analyze the local asymptotics of dimer model correlation functions on tower graphs, and confirm in this case the prediction ([1]) that they converge to those of translation invariant Gibbs measures. In Chapter 4, we construct a Markov chain generalizing domino shuffling which samples exactly from a recently introduced probability measure on tuples of interacting dimer configurations. Exact sampling is extraordinarily useful for the discovery and numerical investigation of asymptotic phenomena in new models. In Chapter 5, we utilize local moves for a different purpose; we construct deterministic tembeddings, which are embeddings of a bipartite graph that are compatible with the underlying 3dimer model. It was recently shown ([2], [3]) that a certain subclass of these, perfect t-embeddings, can be ultimately used to prove “conformal invariance of the model” in the scaling limit. Furthermore, for each local move in the dimer model, there is a corresponding local geometric transformation of t-embeddings ([4]). For Aztec diamond and tower graphs, this allows for an inductive construction of perfect t-embeddings. We utilize the “exact solvability” of the resulting recurrence relations to give exact formulas for the embeddings. We then precisely characterize the global and local asymptotic behavior of the embeddings, and to confirm predictions of [3], [5] in these two cases. In Chapter 6, we utilize the Yang–Baxter equation for a colored generalization of the six-vertex model to compute stationary measures for colored interacting particle systems. In several cases, we match our constructions to existing stationary measures, while in other cases we obtain new stationary measures. We provide a new, unified construction and method of proof (of stationarity) for several different interacting particle systems.