122 research outputs found
Quenched Voronoi percolation
We prove that the probability of crossing a large square in quenched Voronoi
percolation converges to 1/2 at criticality, confirming a conjecture of
Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched
version of the box-crossing property for Voronoi percolation at criticality,
and an Efron-Stein type bound on the variance of the probability of the
crossing event in terms of the sum of the squares of the influences. As a
corollary of the proof, we moreover obtain that the quenched crossing event at
criticality is almost surely noise sensitive.Comment: 21 pages, 2 figure
A Solidification Phenomenon in Random Packings
We prove that uniformly random packings of copies of a certain
simply-connected figure in the plane exhibit global connectedness at all
sufficiently high densities, but not at low densities
Percolation and coarse conformal uniformization
We formulate conjectures regarding percolation on planar triangulations
suggested by assuming (quasi) invariance under coarse conformal uniformization
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