## Duke Mathematical Journal

### 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, $\operatorname{QLE}(\gamma^{2},\eta)$. $\operatorname{QLE}$ 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 $\operatorname{QLE}(2,1)$ as a scaling limit for DLA on a random spanning-tree-decorated planar map and $\operatorname{QLE}(8/3,0)$ as a scaling limit for the Eden model on a random triangulation. We propose using $\operatorname{QLE}(8/3,0)$ to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of $\operatorname{QLE}(8/3,0)$, up to a fixed time, as a metric ball in a random metric space.

#### Article information

Source
Duke Math. J., Volume 165, Number 17 (2016), 3241-3378.

Dates
Revised: 28 October 2015
First available in Project Euclid: 24 October 2016

https://projecteuclid.org/euclid.dmj/1477327539

Digital Object Identifier
doi:10.1215/00127094-3627096

Mathematical Reviews number (MathSciNet)
MR3572845

Zentralblatt MATH identifier
1364.82023

#### Citation

Miller, Jason; Sheffield, Scott. Quantum Loewner evolution. Duke Math. J. 165 (2016), no. 17, 3241--3378. doi:10.1215/00127094-3627096. https://projecteuclid.org/euclid.dmj/1477327539

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