562 research outputs found
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
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
Wiener Index and Remoteness in Triangulations and Quadrangulations
Let be a a connected graph. The Wiener index of a connected graph is the
sum of the distances between all unordered pairs of vertices. We provide
asymptotic formulae for the maximum Wiener index of simple triangulations and
quadrangulations with given connectivity, as the order increases, and make
conjectures for the extremal triangulations and quadrangulations based on
computational evidence. If denotes the arithmetic mean
of the distances from to all other vertices of , then the remoteness of
is defined as the largest value of over all vertices
of . We give sharp upper bounds on the remoteness of simple
triangulations and quadrangulations of given order and connectivity
A bijection for rooted maps on general surfaces
We extend the Marcus-Schaeffer bijection between orientable rooted bipartite
quadrangulations (equivalently: rooted maps) and orientable labeled one-face
maps to the case of all surfaces, that is orientable and non-orientable as
well. This general construction requires new ideas and is more delicate than
the special orientable case, but it carries the same information. In
particular, it leads to a uniform combinatorial interpretation of the counting
exponent for both orientable and non-orientable rooted
connected maps of Euler characteristic , and of the algebraicity of their
generating functions, similar to the one previously obtained in the orientable
case via the Marcus-Schaeffer bijection. It also shows that the renormalization
factor for distances between vertices is universal for maps on all
surfaces: the renormalized profile and radius in a uniform random pointed
bipartite quadrangulation on any fixed surface converge in distribution when
the size tends to infinity. Finally, we extend the Miermont and
Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our
construction opens the way to the study of Brownian surfaces for any compact
2-dimensional manifold.Comment: v2: 55 pages, 22 figure
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