59 research outputs found
Some results on the statistics of hull perimeters in large planar triangulations and quadrangulations
The hull perimeter at distance d in a planar map with two marked vertices at
distance k from each other is the length of the closed curve separating these
two vertices and lying at distance d from the first one (d<k). We study the
statistics of hull perimeters in large random planar triangulations and
quadrangulations as a function of both k and d. Explicit expressions for the
probability density of the hull perimeter at distance d, as well as for the
joint probability density of hull perimeters at distances d1 and d2, are
obtained in the limit of infinitely large k. We also consider the situation
where the distance d at which the hull perimeter is measured corresponds to a
finite fraction of k. The various laws that we obtain are identical for
triangulations and for quadrangulations, up to a global rescaling. Our approach
is based on recursion relations recently introduced by the author which
determine the generating functions of so-called slices, i.e. pieces of maps
with appropriate distance constraints. It is indeed shown that the map
decompositions underlying these recursion relations are intimately linked to
the notion of hull perimeters and provide a simple way to fully control them.Comment: 32 pages, 16 figure
The three-point function of general planar maps
We compute the distance-dependent three-point function of general planar maps
and of bipartite planar maps, i.e., the generating function of these maps with
three marked vertices at prescribed pairwise distances. Explicit expressions
are given for maps counted by their number of edges only, or by both their
numbers of edges and faces. A few limiting cases and applications are
discussed.Comment: 33 pages, 12 figure
Monte Carlo Results for Projected Self-Avoiding Polygons: A Two-dimensional Model for Knotted Polymers
We introduce a two-dimensional lattice model for the description of knotted
polymer rings. A polymer configuration is modeled by a closed polygon drawn on
the square diagonal lattice, with possible crossings describing pairs of
strands of polymer passing on top of each other. Each polygon configuration can
be viewed as the two- dimensional projection of a particular knot. We study
numerically the statistics of large polygons with a fixed knot type, using a
generalization of the BFACF algorithm for self-avoiding walks. This new
algorithm incorporates both the displacement of crossings and the three types
of Reidemeister transformations preserving the knot topology. Its ergodicity
within a fixed knot type is not proven here rigorously but strong arguments in
favor of this ergodicity are given together with a tentative sketch of proof.
Assuming this ergodicity, we obtain numerically the following results for the
statistics of knotted polygons: In the limit of a low crossing fugacity, we
find a localization along the polygon of all the primary factors forming the
knot. Increasing the crossing fugacity gives rise to a transition from a
self-avoiding walk to a branched polymer behavior.Comment: 36 pages, 30 figures, latex, epsf. to appear in J.Phys.A: Math. Ge
The distance-dependent two-point function of quadrangulations: a new derivation by direct recursion
22 pages, 14 figuresInternational audienceWe give a new derivation of the distance-dependent two-point function of planar quadrangulations by solving a new direct recursion relation for the associated slice generating functions. Our approach for both the derivation and the solution of this new recursion is in all points similar to that used recently by the author in the context of planar triangulations
Eulerian triangulations: two-point function and hull perimeter statistics
30 pages, 17 figuresInternational audienceWe present a new derivation of the distance-dependent two-point function for planar Eulerian triangulations and give expressions for more refined generating functions where we also control hull perimeters. These results are obtained in the framework of a new recursion relation for slice generating functions and extend similar results obtained recently for triangulations and quadrangulations. A number of explicit formulas are given for the statistics of hull perimeters in infinitely large random planar Eulerian triangulations
The distance-dependent two-point function of triangulations: a new derivation from old results
International audienceWe present a new derivation of the distance-dependent two-point function of random planar triangulations. As it is well-known, this function is intimately related to the generating functions of so-called slices, which are pieces of triangulation having boundaries made of shortest paths of prescribed length. We show that the slice generating functions are fully determined by a direct recursive relation on their boundary length. Remarkably, the kernel of this recursion is some quantity introduced and computed by Tutte a long time ago in the context of a global enumeration of planar triangulations. We may thus rely on these old results to solve our new recursion relation explicitly in a constructive way
On the two-point function of general planar maps and hypermaps
We consider the problem of computing the distance-dependent two-point
function of general planar maps and hypermaps, i.e. the problem of counting
such maps with two marked points at a prescribed distance. The maps considered
here may have faces of arbitrarily large degree, which requires new bijections
to be tackled. We obtain exact expressions for the following cases: general and
bipartite maps counted by their number of edges, 3-hypermaps and
3-constellations counted by their number of dark faces, and finally general and
bipartite maps counted by both their number of edges and their number of faces.Comment: 32 pages, 17 figure
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