Duke Mathematical Journal

Quantum Loewner evolution

Jason Miller and Scott Sheffield

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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 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.

Article information

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

Dates
Received: 3 July 2014
Revised: 28 October 2015
First available in Project Euclid: 24 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1477327539

Digital Object Identifier
doi:10.1215/00127094-3627096

Mathematical Reviews number (MathSciNet)
MR3572845

Zentralblatt MATH identifier
1364.82023

Subjects
Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 82B24: Interface problems; diffusion-limited aggregation

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

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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