Phase Transitions and Scaling Limits in Lattice Models

University Of Connecticut, Storrs CT

Investigators

Abstract

Phase transitions, ranging from familiar occurrences such as liquid water freezing into ice to technologically critical phenomena such as transition to superconductivity, are ubiquitous throughout nature, have been an active topic in mathematical and physical research for as long as science has existed, and continue to pose important research challenges. Lattice models, constructed to study natural phase transitions, are probability distributions on subsets of graphs depending on continuous parameters. This research project will study the phase transition of several important lattice models, including (1) the constrained percolation model, which randomly assigns "0" or "1" to each vertex of a graph, satisfying constraints that represent aspects of molecular structure; and (2) the self-avoiding walk, which is a path on a graph visiting each vertex at most once, introduced as a model for long-chain polymers. The work will broaden and deepen the mathematical foundations of the subject, with potential important applications in theoretical physics and chemistry. The investigator aims to develop new theory concerning the phase transition of certain lattice models. Imposing constraints in the percolation model usually makes the model lose stochastic monotonicity, which, in the unconstrained case, is critical to study the phase transition described by the behavior of infinite clusters. One goal of the project is to develop new combinatorial and probabilistic techniques to study the behavior of infinite clusters without using stochastic monotonicity. Enumerating self-avoiding walks is typically difficult due to the non-Markovian structure. The connective constant and exponent of self-avoiding walks are fundamental properties of the underlying graph, yet can be identified for very few graphs. Another goal of this project is to obtain new information about connective constants and exponents for large classes of graphs, which may be achieved by analyzing harmonic functions on graphs. An analog to the central limit theorem in two dimensions is the limit shape behavior of height functions of certain lattice models at criticality, i.e. when the phase transition occurs. The investigator also plans to study the limit shape, using techniques from complex analysis and algebraic geometry.