456 research outputs found
Combinatorics of Hard Particles on Planar Graphs
We revisit the problem of hard particles on planar random tetravalent graphs
in view of recent combinatorial techniques relating planar diagrams to
decorated trees. We show how to recover the two-matrix model solution to this
problem in this purely combinatorial language.Comment: 35 pages, 20 figures, tex, harvmac, eps
Distance statistics in large toroidal maps
We compute a number of distance-dependent universal scaling functions
characterizing the distance statistics of large maps of genus one. In
particular, we obtain explicitly the probability distribution for the length of
the shortest non-contractible loop passing via a random point in the map, and
that for the distance between two random points. Our results are derived in the
context of bipartite toroidal quadrangulations, using their coding by
well-labeled 1-trees, which are maps of genus one with a single face and
appropriate integer vertex labels. Within this framework, the distributions
above are simply obtained as scaling limits of appropriate generating functions
for well-labeled 1-trees, all expressible in terms of a small number of basic
scaling functions for well-labeled plane trees.Comment: 24 pages, 9 figures, minor corrections, new added reference
Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof
We consider the problem of enumeration of planar maps and revisit its
one-matrix model solution in the light of recent combinatorial techniques
involving conjugated trees. We adapt and generalize these techniques so as to
give an alternative and purely combinatorial solution to the problem of
counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps
Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
We consider quadrangulations with a boundary and derive explicit expressions
for the generating functions of these maps with either a marked vertex at a
prescribed distance from the boundary, or two boundary vertices at a prescribed
mutual distance in the map. For large maps, this yields explicit formulas for
the bulk-boundary and boundary-boundary correlators in the various encountered
scaling regimes: a small boundary, a dense boundary and a critical boundary
regime. The critical boundary regime is characterized by a one-parameter family
of scaling functions interpolating between the Brownian map and the Brownian
Continuum Random Tree. We discuss the cases of both generic and self-avoiding
boundaries, which are shown to share the same universal scaling limit. We
finally address the question of the bulk-loop distance statistics in the
context of planar quadrangulations equipped with a self-avoiding loop. Here
again, a new family of scaling functions describing critical loops is
discovered.Comment: 55 pages, 14 figures, final version with minor correction
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
Geodesic Distance in Planar Graphs
We derive the exact generating function for planar maps (genus zero
fatgraphs) with vertices of arbitrary even valence and with two marked points
at a fixed geodesic distance. This is done in a purely combinatorial way based
on a bijection with decorated trees, leading to a recursion relation on the
geodesic distance. The latter is solved exactly in terms of discrete
soliton-like expressions, suggesting an underlying integrable structure. We
extract from this solution the fractal dimensions at the various
(multi)-critical points, as well as the precise scaling forms of the continuum
two-point functions and the probability distributions for the geodesic distance
in (multi)-critical random surfaces. The two-point functions are shown to obey
differential equations involving the residues of the KdV hierarchy.Comment: 38 pages, 8 figures, tex, harvmac, eps
Planar maps and continued fractions
We present an unexpected connection between two map enumeration problems. The
first one consists in counting planar maps with a boundary of prescribed
length. The second one consists in counting planar maps with two points at a
prescribed distance. We show that, in the general class of maps with controlled
face degrees, the solution for both problems is actually encoded into the same
quantity, respectively via its power series expansion and its continued
fraction expansion. We then use known techniques for tackling the first problem
in order to solve the second. This novel viewpoint provides a constructive
approach for computing the so-called distance-dependent two-point function of
general planar maps. We prove and extend some previously predicted exact
formulas, which we identify in terms of particular Schur functions.Comment: 47 pages, 17 figures, final version (very minor changes since v2
Statistics of geodesics in large quadrangulations
We study the statistical properties of geodesics, i.e. paths of minimal
length, in large random planar quadrangulations. We extend Schaeffer's
well-labeled tree bijection to the case of quadrangulations with a marked
geodesic, leading to the notion of "spine trees", amenable to a direct
enumeration. We obtain the generating functions for quadrangulations with a
marked geodesic of fixed length, as well as with a set of "confluent
geodesics", i.e. a collection of non-intersecting minimal paths connecting two
given points. In the limit of quadrangulations with a large area n, we find in
particular an average number 3*2^i of geodesics between two fixed points at
distance i>>1 from each other. We show that, for generic endpoints, two
confluent geodesics remain close to each other and have an extensive number of
contacts. This property fails for a few "exceptional" endpoints which can be
linked by truly distinct geodesics. Results are presented both in the case of
finite length i and in the scaling limit i ~ n^(1/4). In particular, we give
the scaling distribution of the exceptional points.Comment: 37 pages, 18 color figures, improved version with several
clarifications (mostly in sections 2.1 and 2.4) and one added section (3.1)
on ensembles of random quadrangulation
Unicyclic Components in Random Graphs
The distribution of unicyclic components in a random graph is obtained
analytically. The number of unicyclic components of a given size approaches a
self-similar form in the vicinity of the gelation transition. At the gelation
point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a
result, the total number of unicyclic components grows logarithmically with the
system size.Comment: 4 pages, 2 figure
Uniform infinite planar triangulation and related time-reversed critical branching process
We establish a connection between the uniform infinite planar triangulation
and some critical time-reversed branching process. This allows to find a
scaling limit for the principal boundary component of a ball of radius R for
large R (i.e. for a boundary component separating the ball from infinity). We
show also that outside of R-ball a contour exists that has length linear in R.Comment: 27 pages, 5 figures, LaTe
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