373 research outputs found
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
We study Markov chains for -orientations of plane graphs, these are
orientations where the outdegree of each vertex is prescribed by the value of a
given function . The set of -orientations of a plane graph has
a natural distributive lattice structure. The moves of the up-down Markov chain
on this distributive lattice corresponds to reversals of directed facial cycles
in the -orientation. We have a positive and several negative results
regarding the mixing time of such Markov chains.
A 2-orientation of a plane quadrangulation is an orientation where every
inner vertex has outdegree 2. We show that there is a class of plane
quadrangulations such that the up-down Markov chain on the 2-orientations of
these quadrangulations is slowly mixing. On the other hand the chain is rapidly
mixing on 2-orientations of quadrangulations with maximum degree at most 4.
Regarding examples for slow mixing we also revisit the case of 3-orientations
of triangulations which has been studied before by Miracle et al.. Our examples
for slow mixing are simpler and have a smaller maximum degree, Finally we
present the first example of a function and a class of plane
triangulations of constant maximum degree such that the up-down Markov chain on
the -orientations of these graphs is slowly mixing
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
3D Lorentzian Quantum Gravity from the asymmetric ABAB matrix model
The asymmetric ABAB-matrix model describes the transfer matrix of
three-dimensional Lorentzian quantum gravity. We study perturbatively the
scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution
and deduce the associated renormalization of three-dimensional Lorentzian
quantum gravity.Comment: 21 pages, typo in references correcte
Simple recurrence formulas to count maps on orientable surfaces
We establish a simple recurrence formula for the number of rooted
orientable maps counted by edges and genus. We also give a weighted variant for
the generating polynomial where is a parameter taking the number
of faces of the map into account, or equivalently a simple recurrence formula
for the refined numbers that count maps by genus, vertices, and
faces. These formulas give by far the fastest known way of computing these
numbers, or the fixed-genus generating functions, especially for large . In
the very particular case of one-face maps, we recover the Harer-Zagier
recurrence formula.
Our main formula is a consequence of the KP equation for the generating
function of bipartite maps, coupled with a Tutte equation, and it was
apparently unnoticed before. It is similar in look to the one discovered by
Goulden and Jackson for triangulations, and indeed our method to go from the KP
equation to the recurrence formula can be seen as a combinatorial
simplification of Goulden and Jackson's approach (together with one additional
combinatorial trick). All these formulas have a very combinatorial flavour, but
finding a bijective interpretation is currently unsolved.Comment: Version 3: We changed the title once again. We also corrected some
misprints, gave another equivalent formulation of the main result in terms of
vertices and faces (Thm. 5), and added complements on bivariate generating
functions. Version 2: We extended the main result to include the ability to
track the number of faces. The title of the paper has been changed
accordingl
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
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