10 research outputs found
Vertex Degrees in Planar Maps
We prove a general multi-dimensional central limit theorem for the expected
number of vertices of a given degree in the family of planar maps whose vertex
degrees are restricted to an arbitrary (finite or infinite) set of positive
integers D. Our results rely on a classical bijection with mobiles (objects
exhibiting a tree structure), combined with refined analytic tools to deal with
the systems of equations on infinite variables that arise. We also discuss some
possible extension to maps of higher genus.Comment: 16 pages typos removed, author's name correcte
Pattern occurrences in random planar maps
We consider planar maps adjusted with a (regular critical) Boltzmann
distribution and show that the expected number of pattern occurrences of a
given map is asymptotically linear when the number n of edges goes to infinity.
The main ingredient for the proof is an extension of a formula by Liskovets
(1999)
The evolution of random graphs on surfaces
For integers and , let denote the graph taken
uniformly at random from the set of all graphs on with
exactly edges and with genus at most . We use counting arguments to
investigate the components, subgraphs, maximum degree, and largest face size of
, finding that there is often different asymptotic behaviour
depending on the ratio .
In our main results, we show that the probability that contains
any given non-planar component converges to as for all
; the probability that contains a copy of any given planar
graph converges to as if ; the
maximum degree of is with high probability if
; and the largest face size of has a
threshold around where it changes from to with high probability.Comment: 35 page
Expected number of pattern and submap occurrences in random planar maps
Drmota and Stufler proved recently that the expected number of pattern
occurrences of a given map is asymptotically linear when the number of edges
goes to infinity. In this paper we improve their result by means of a different
method. Our method allows us to develop a systematic way for computing the
explicit constant of the linear (main) term and shows that it is a positive
rational number. Moreover, by extending our method, we also solve the
corresponding problem of submap occurrences.Comment: 21 pages, 7 figure
A Markov Chain Sampler for Plane Curves
A plane curve is a knot diagram in which each crossing is replaced by a
4-valent vertex, and so are dual to a subset of planar quadrangulations. The
aim of this paper is to introduce a new tool for sampling diagrams via sampling
of plane curves. At present the most efficient method for sampling diagrams is
rejection sampling, however that method is inefficient at even modest sizes. We
introduce Markov chains that sample from the space of plane curves using local
moves based on Reidemeister moves. By then mapping vertices on those curves to
crossings we produce random knot diagrams. Combining this chain with flat
histogram methods we achieve an efficient sampler of plane curves and knot
diagrams. By analysing data from this chain we are able to estimate the number
of knot diagrams of a given size and also compute knotting probabilities and so
investigate their asymptotic behaviour.Comment: 41 pages, 30 figure
Limit Laws of Planar Maps with Prescribed Vertex Degrees
We prove a general multi-dimensional central limit theorem for the expected
number of vertices of a given degree in the family of planar maps whose vertex
degrees are restricted to an arbitrary (finite or infinite) set of positive
integers . Our results rely on a classical bijection with mobiles (objects
exhibiting a tree structure), combined with refined analytic tools to deal with
the systems of equations on infinite variables that arise. We also discuss
possible extensions to maps of higher genus and to weighted maps.Comment: 22 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1605.0420
Universal singular exponents in catalytic variable equations
Catalytic equations appear in several combinatorial applications, most
notably in the numeration of lattice path and in the enumeration of planar
maps. The main purpose of this paper is to show that the asymptotic estimate
for the coefficients of the solutions of (so-called) positive catalytic
equations has a universal asymptotic behavior. In particular, this provides a
rationale why the number of maps of size in various planar map classes
grows asymptotically like , for suitable positive
constants and . Essentially we have to distinguish between linear
catalytic equations (where the subexponential growth is ) and
non-linear catalytic equations (where we have as in planar maps).
Furthermore we provide a quite general central limit theorem for parameters
that can be encoded by catalytic functional equations, even when they are not
positive.Comment: 21 page
A Central Limit Theorem for the Number of Degree-k Vertices in Random Maps
We prove that the number of vertices of given degree in random planar maps satisfies a central limit theorem with mean and variance that are asymptotically linear in the number of edges. The proof relies on an analytic version of the quadratic method and singularity analysis of multivariate generating functions.
Local convergence of random planar graphs
The present work describes the asymptotic local shape of a graph drawn
uniformly at random from all connected simple planar graphs with n labelled
vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched
limit in the local topology as n tends to infinity. We also establish such
limits for random 2-connected planar graphs and maps as their number of edges
tends to infinity. Our approach encompasses a new probabilistic view on the
Tutte decomposition. This allows us to follow the path along the decomposition
of connectivity from planar maps to planar graphs in a uniformed way, basing
each step on condensation phenomena for random walks under subexponentiality
and Gibbs partitions. Using large deviation results, we recover the asymptotic
formula by Gim\'enez and Noy (2009) for the number of planar graphs
Algorithmica manuscript No. (will be inserted by the editor) A Central Limit Theorem for the Number of Degree-k Vertices in Random Maps
Abstract We prove that the number of vertices of given degree in (general or 2-connected) random planar maps satisfies a central limit theorem with mean and variance that are asymptotically linear in the number of edges. The proof relies on an analytic version of the quadratic method and singularity analysis of multivariate generating functions