276 research outputs found
Percolation on uniform infinite planar maps
We construct the uniform infinite planar map (UIPM), obtained as the n \to
\infty local limit of planar maps with n edges, chosen uniformly at random. We
then describe how the UIPM can be sampled using a "peeling" process, in a
similar way as for uniform triangulations. This process allows us to prove that
for bond and site percolation on the UIPM, the percolation thresholds are
p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other
classes of random infinite planar maps, and we show in particular that for bond
percolation on the uniform infinite planar quadrangulation, the percolation
threshold is p_c^bond=1/3.Comment: 26 pages, 9 figure
Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary
We show that all geodesic rays in the uniform infinite half-planar
quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering
thereby a recent question of Curien. However, the possible intersection points
are sparsely distributed along the boundary. As an intermediate step, we show
that geodesic rays in the UIHPQ are proper, a fact that was recently
established by Caraceni and Curien (2015) by a reasoning different from ours.
Finally, we argue that geodesic rays in the uniform infinite half-planar
triangulation behave in a very similar manner, even in a strong quantitative
sense.Comment: 29 pages, 13 figures. Added reference and figur
A view from infinity of the uniform infinite planar quadrangulation
We introduce a new construction of the Uniform Infinite Planar
Quadrangulation (UIPQ). Our approach is based on an extension of the
Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the
spirit of previous work. However, we release the positivity constraint on the
labels of trees which was imposed in these references, so that our construction
is technically much simpler. This approach allows us to prove the conjectures
of Krikun pertaining to the "geometry at infinity" of the UIPQ, and to derive
new results about the UIPQ, among which a fine study of infinite geodesics.Comment: 39 pages, 11 figure
The two uniform infinite quadrangulations of the plane have the same law
We prove that the uniform infinite random quadrangulations defined
respectively by Chassaing-Durhuus and Krikun have the same distribution.Comment: English version of arXiv:0805.4687, with various improvement
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