Presented by:
Joshua Pfeffer
Date:
Friday 20th July 2018 - 13:45 to 14:05
Venue:
INI Seminar Room 1
Event:
Abstract:
External
diffusion limited aggregation (DLA) is a widely studied
subject in the physics literature, with many manifestations
in nature; but it is
not well-understood mathematically in any environment. We
consider external
DLA on an infinite spanning-tree-weighted random planar
map. We prove that the
growth exponent for the external diameter of the DLA
cluster exists and is equal
to $2/d _{\sqrt{2}}$, where $d_{\sqrt{2}}$ denotes the
``fractal dimension
of $\sqrt{2}$-Liouville quantum gravity (LQG)''---or,
equivalently, the ball
volume growth exponent for the spanning-tree weighted map.
Our proof is based
on the fact that the complement of an external DLA cluster
on a spanning-tree
weighted map is a spanning-tree weighted map with boundary,
which allows
us to reduce our problem to proving certain estimates for
distances in random
planar maps with boundary. This is joint work with Ewain
Gwynne.
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