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 $\eta$-DBM, a generalization of DLA in which particle locations are sampled from the $\eta$th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $\eta$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $\gamma \in [0,2]$.

In this generality, we propose a scaling limit candidate called quantum Loewner evolution, $QLE({\gamma }^2,\eta )$. $QLE$ is defined in terms of the radial Loewner equation like radial stochastic Loewner evolution, except that it is driven by a measure-valued diffusion ${\nu }_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ${\nu }_t$ using a stochastic partial differential equation. For each $\gamma \in (0,2]$, there are two or three special values of $\eta$ for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ${\nu }_t$.

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.