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.

  • 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:

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

Jason Miller and Scott Sheffield Advance publication (2016)

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 modelη-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.