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

A note on irreducible maps with several boundaries

We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4). Our derivation is based on a tree interpretation of the various encountered generating functions.Comment: 18 pages, 8 figure

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