15 November 2016 Quantum Loewner evolution
Jason Miller, Scott Sheffield
Duke Math. J. 165(17): 3241-3378 (15 November 2016). DOI: 10.1215/00127094-3627096


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 ηth 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[0,2].

In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ2,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 νt derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of νt using a stochastic partial differential equation. For each γ(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 νt.

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.


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Jason Miller. Scott Sheffield. "Quantum Loewner evolution." Duke Math. J. 165 (17) 3241 - 3378, 15 November 2016. https://doi.org/10.1215/00127094-3627096


Received: 3 July 2014; Revised: 28 October 2015; Published: 15 November 2016
First available in Project Euclid: 24 October 2016

zbMATH: 1364.82023
MathSciNet: MR3572845
Digital Object Identifier: 10.1215/00127094-3627096

Primary: 82B41
Secondary: 82B24

Keywords: Brownian map , dielectric-breakdown mode , Diffusion-limited aggregation , Gaussian free field , Liouville quantum gravity , Schramm–Loewner evolution

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 17 • 15 November 2016
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