Gaussian Free Field and Conformal Loop Ensemble

This research is in the area of Probability and Statistical Physics. Statistical physicists and probabilists often try to understand the macroscopic behavior of systems consisting of many microscopic random inputs, which can give rise to interfaces between two phases at a critical temperature, such as water and ice at zero degree Celsius. This can be modeled via the scaling limit behavior (macroscopic behavior) of discrete lattice models (microscopic inputs) in critical cases. Oded Schramm's SLE (Stochastic Loewner Evolution) processes have led mathematicians and physicists to a clean and novel understanding of the scaling limits of the interfaces in discrete models in two dimensions. And CLE (Conformal Loop Ensemble) is the generalization of SLE which is predicted to be the scaling limit of the collection of all interfaces in discrete models. This research focuses on CLE and its relation between Gaussian Free Field, Liouville Quantum Gravity, and Random Maps.

Precisely, the research considers the following two problems.

(1) The conformally invariant metric on CLE. Since the introduction of SLE and CLE, CLE(4) has been proved to be the scaling limit of the collection of level lines of discrete Gaussian Free Field. In the previous work of the principal investigator, a time parameter is constructed for CLE(4) loop configurations, and a coupling between Gaussian Free Field and CLE(4) with time parameter is given. The research aims to show that the time parameter defined on CLE(4) is in fact a deterministic function of the loop configuration. This result would deepen understanding of the relation between Gaussian Free Field and CLE.

(2) CLE-decorated Liouville Quantum Gravity. Liouville Quantum Gravity is conjectured to be the scaling limit of random maps. The research project first aims to understand CLE-decorated Liouville Quantum Gravity and then to explore the relation between CLE-decorated Liouville Quantum Gravity and loop-decorated random maps.