Quantum Loewner evolution

Abstract

What is the scaling limit of diffusion-limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to conve rge in law to a Liouville quantum gravity (LQG) surface with parameter γ e [0,2]. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 ,η). QLE is defined in terms of the radial Loewner equation like radial stochastic Loewner evolution, except that it is driven by a measure-valued diffusion v t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of v t using a stochastic partial differential equation. For each γ e [0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of v t . We also explain discrete versions of our construction that relate DLA to loop-erased random walks and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot-machine reels) facilitates explicit calculation. We propose QLE(2, 1) as a scaling limit for DLA on a random spanning-tree-decorated planar map and QLE(8/3, 0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3, 0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3, 0), up to a fixed time, as a metric ball in a random metric space.