17,012 research outputs found
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
Local limit of labeled trees and expected volume growth in a random quadrangulation
Exploiting a bijective correspondence between planar quadrangulations and
well-labeled trees, we define an ensemble of infinite surfaces as a limit of
uniformly distributed ensembles of quadrangulations of fixed finite volume. The
limit random surface can be described in terms of a birth and death process and
a sequence of multitype Galton--Watson trees. As a consequence, we find that
the expected volume of the ball of radius around a marked point in the
limit random surface is .Comment: Published at http://dx.doi.org/10.1214/009117905000000774 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Rayleigh processes, real trees, and root growth with re-grafting
The real trees form a class of metric spaces that extends the class of trees
with edge lengths by allowing behavior such as infinite total edge length and
vertices with infinite branching degree. Aldous's Brownian continuum random
tree, the random tree-like object naturally associated with a standard Brownian
excursion, may be thought of as a random compact real tree. The continuum
random tree is a scaling limit as N tends to infinity of both a critical
Galton-Watson tree conditioned to have total population size N as well as a
uniform random rooted combinatorial tree with N vertices. The Aldous--Broder
algorithm is a Markov chain on the space of rooted combinatorial trees with N
vertices that has the uniform tree as its stationary distribution. We construct
and study a Markov process on the space of all rooted compact real trees that
has the continuum random tree as its stationary distribution and arises as the
scaling limit as N tends to infinity of the Aldous--Broder chain. A key
technical ingredient in this work is the use of a pointed Gromov--Hausdorff
distance to metrize the space of rooted compact real trees.Comment: 48 Pages. Minor revision of version of Feb 2004. To appear in
Probability Theory and Related Field
- …