On Multiple SLE Systems and their Deterministic Limits

Citation

Zhang, Jiaxin (2025) On Multiple SLE Systems and their Deterministic Limits. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/spf7-9j65. https://resolver.caltech.edu/CaltechTHESIS:05202025-052235420

Abstract

In this thesis, we study multiple radial SLE(k) systems -- a family of random multi-curve systems in a simply-connected domain Ω, with marked boundary points z₁....zₙ \in ∂Ω and a marked interior point q, where parameter k > 0 measures the randomness of the system. We also study the multiple radial SLE(0) systems as the deterministic limit of multiple radial SLE(k) systems.

As a consequence of domain Markov property and conformal invariance, we derive that a multiple radial SLE(k) system is characterized by a conformally covariant partition function satisfying the null vector equations--a second-order PDE system. On the other hand, using the Coulomb gas method inspired by conformal field theory, we construct four types of solutions to the null vector equations, which can be classified according to topological link patterns.

We construct the multiple radial SLE(0) systems from stationary relations by heuristically taking the classical limit of partition functions as k > 0. By constructing the field integrals of motion for the Loewner dynamics, we show that the traces of multiple radial SLE(0) systems are the horizontal trajectories of an equivalence class of quadratic differentials. These trajectories have limiting ends at the growth points and form a radial link pattern.

The stationary relations connect the classification of multiple radial SLE(0) systems to the enumeration of critical points of the master function of trigonometric Knizhnik-Zamolodchikov (KZ) equations.

For k > 0$, the partition functions of multiple radial SLE(k) systems correspond to eigenstates of the quantum Calogero-Sutherland (CS) Hamiltonian beyond the fermionic states. In the deterministic case of k=0, we show that the Loewner dynamics with a common parametrization of capacity form a special class of classical CS systems, restricted to a submanifold of phase space defined by the Lax matrix.

Thesis Files