## Duke Mathematical Journal

Published by Duke University Press since its inception in 1935, the *Duke Mathematical Journal* is one of the world's leading mathematical journals. *DMJ* emphasizes the most active and influential areas of current mathematics. Advance publication of articles online is available.

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**ISSN:**0012-7094 (print), 1547-7398 (electronic)**Publisher:**Duke University Press**Discipline(s):**Mathematics**Full text available in Euclid:**1935--**Access:**By subscription only**Euclid URL:**http://projecteuclid.org/dmj

### Featured bibliometrics

*MR Citation Database* MCQ (2015): 2.29

*JCR* (2015) Impact Factor: 2.350

*JCR* (2015) Five-year Impact Factor: 2.337

*JCR* (2015) Ranking: 9/312 (Mathematics)

Article Influence: 3.899

Eigenfactor: Duke Mathematical Journal

SJR/SCImago Journal Rank (2015): 5.675

Indexed/Abstracted in: *CompuMath Citation Index, Current Contents: Physical, Chemical, and Earth Sciences, International Bibliography of Periodical Literature (IBZ), ISI Science Citation Index Expanded, Magazines for Libraries, MathSciNet, Scopus, zbMATH*

### Featured article

* *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.