arXiv2013
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 {\em 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 converge in law to a Liouville quantum gravity (LQG) surface with parameter $γ\in [0,2]$. 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 SLE, except that it is driven by a measure valued diffusion $ν_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of $ν_t$ using an SPDE. For each $γ\in (0,2]$, 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 $ν_t$. We also explain discrete versions of our construction that relate DLA to loop-erased random walk 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.