615 research outputs found
Scaling Limits for Internal Aggregation Models with Multiple Sources
We study the scaling limits of three different aggregation models on Z^d:
internal DLA, in which particles perform random walks until reaching an
unoccupied site; the rotor-router model, in which particles perform
deterministic analogues of random walks; and the divisible sandpile, in which
each site distributes its excess mass equally among its neighbors. As the
lattice spacing tends to zero, all three models are found to have the same
scaling limit, which we describe as the solution to a certain PDE free boundary
problem in R^d. In particular, internal DLA has a deterministic scaling limit.
We find that the scaling limits are quadrature domains, which have arisen
independently in many fields such as potential theory and fluid dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in
v2: added "least action principle" (Lemma 3.2); small corrections in section
4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version);
expanded section 6.
Hele-Shaw flow on weakly hyperbolic surfaces
We consider the Hele-Shaw flow that arises from injection of two-dimensional
fluid into a point of a curved surface. The resulting fluid domains have and
are more or less determined implicitly by a mean value property for harmonic
functions. We improve on the results of Hedenmalm and Shimorin \cite{HS} and
obtain essentially the same conclusions while imposing a weaker curvature
condition on the surface. Incidentally, the curvature condition is the same as
the one that appears in a recent paper of Hedenmalm and Perdomo, where the
problem of finding smooth area minimizing surfaces for a given curvature form
under a natural normalizing condition was considered. Probably there are deep
reasons behind this coincidence.Comment: 16 page
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