Random Voronoi Percolation in the Plane
Phase Transitions in Computation and Reconstruction
March 08, 2005 04:00 PM to 04:40 PM
Speakers:
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Abstract: |
Let $\Lambda \subset {\mathbb R}^2$ be the set of points of a Poisson
process of intensity 1 in the plane; the sets $V(z)=\{x\in {\mathbb R}^2: d
(x,z)\le d(x,y) \ \text{for all} \ y\in \Lambda \}$, $z\in \Lambda$, are the {\em
Voronoi cells} with respect to $\Lambda$. Colour each cell black with
probability $p$, independently of the other cells (and leave it white
otherwise). Trivially, there is a critical probability $p_c$ such that
if $p>p_c$ then the union of the black cells contains an unbounded
component a.s., while for $p
trivial to conjecture that this critical probability is $1/2$, and a little while
ago Oliver Riordan and I managed to prove this conjecture.
As a bonus, we have also obtained a particularly simple proof of the
Harris-Kesten theorem about the critical probability of bond
percolation on ${\mathbb Z}^2$.
In the talk I shall present some of these recent results. |
Keywords: |
Phase Transition;Computation;Reconstruction |
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