650 research outputs found
Simple maps, Hurwitz numbers, and Topological Recursion
We introduce the notion of fully simple maps, which are maps with non
self-intersecting disjoint boundaries. In contrast, maps where such a
restriction is not imposed are called ordinary. We study in detail the
combinatorics of fully simple maps with topology of a disk or a cylinder. We
show that the generating series of simple disks is given by the functional
inversion of the generating series of ordinary disks. We also obtain an elegant
formula for cylinders. These relations reproduce the relation between moments
and free cumulants established by Collins et al. math.OA/0606431, and implement
the symplectic transformation on the spectral curve in
the context of topological recursion. We conjecture that the generating series
of fully simple maps are computed by the topological recursion after exchange
of and . We propose an argument to prove this statement conditionally to
a mild version of symplectic invariance for the -hermitian matrix model,
which is believed to be true but has not been proved yet.
Our argument relies on an (unconditional) matrix model interpretation of
fully simple maps, via the formal hermitian matrix model with external field.
We also deduce a universal relation between generating series of fully simple
maps and of ordinary maps, which involves double monotone Hurwitz numbers. In
particular, (ordinary) maps without internal faces -- which are generated by
the Gaussian Unitary Ensemble -- and with boundary perimeters
are strictly monotone double Hurwitz numbers
with ramifications above and above .
Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this
implies an ELSV-like formula for these Hurwitz numbers.Comment: 66 pages, 7 figure
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus and a set
of rooted plane trees with distinguished vertices. The bijection applies as
well to the case of labelled unicellular maps, which are related to all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of unicellular
maps of given genus, and a simple bijective proof of a formula of Lehman and
Walsh on the number of triangulations with one vertex. From the labelled case,
we deduce an expression of the asymptotic number of maps of genus g with n
edges involving the ISE random measure, and an explicit characterization of the
limiting profile and radius of random bipartite quadrangulations of genus g in
terms of the ISE.Comment: 27pages, 6 figures, to appear in PTRF. Version 2 includes corrections
from referee report in sections 6-
A look at cycles containing specified elements of a graph
AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration
Large deviations of empirical neighborhood distribution in sparse random graphs
Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is
present independently with probability c/n, with c>0 fixed. For large n, a
typical random graph locally behaves like a Galton-Watson tree with Poisson
offspring distribution with mean c. Here, we study large deviations from this
typical behavior within the framework of the local weak convergence of finite
graph sequences. The associated rate function is expressed in terms of an
entropy functional on unimodular measures and takes finite values only at
measures supported on trees. We also establish large deviations for other
commonly studied random graph ensembles such as the uniform random graph with
given number of edges growing linearly with the number of vertices, or the
uniform random graph with given degree sequence. To prove our results, we
introduce a new configuration model which allows one to sample uniform random
graphs with a given neighborhood distribution, provided the latter is supported
on trees. We also introduce a new class of unimodular random trees, which
generalizes the usual Galton Watson tree with given degree distribution to the
case of neighborhoods of arbitrary finite depth. These generalized Galton
Watson trees turn out to be useful in the analysis of unimodular random trees
and may be considered to be of interest in their own right.Comment: 58 pages, 5 figure
A view from infinity of the uniform infinite planar quadrangulation
We introduce a new construction of the Uniform Infinite Planar
Quadrangulation (UIPQ). Our approach is based on an extension of the
Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the
spirit of previous work. However, we release the positivity constraint on the
labels of trees which was imposed in these references, so that our construction
is technically much simpler. This approach allows us to prove the conjectures
of Krikun pertaining to the "geometry at infinity" of the UIPQ, and to derive
new results about the UIPQ, among which a fine study of infinite geodesics.Comment: 39 pages, 11 figure
Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) without Flat Angles
A foundational result in origami mathematics is Kawasaki and Justin's simple,
efficient characterization of flat foldability for unassigned single-vertex
crease patterns (where each crease can fold mountain or valley) on flat
material. This result was later generalized to cones of material, where the
angles glued at the single vertex may not sum to . Here we
generalize these results to when the material forms a complex (instead of a
manifold), and thus the angles are glued at the single vertex in the structure
of an arbitrary planar graph (instead of a cycle). Like the earlier
characterizations, we require all creases to fold mountain or valley, not
remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly
for flat material and strongly for complexes). Equivalently, we efficiently
characterize which combinatorially embedded planar graphs with prescribed edge
lengths can fold flat, when all angles must be mountain or valley (not unfolded
flat). Our algorithm runs in time, improving on the previous
best algorithm of .Comment: 17 pages, 8 figures, to appear in Proceedings of the 38th
International Symposium on Computational Geometr
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