15 research outputs found
Explicit enumeration of triangulations with multiple boundaries
We enumerate rooted triangulations of a sphere with multiple holes by the
total number of edges and the length of each boundary component. The proof
relies on a combinatorial identity due to W.T. Tutte
A simple formula for the series of constellations and quasi-constellations with boundaries
We obtain a very simple formula for the generating function of bipartite
(resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed
lengths, which generalizes certain expressions obtained by Eynard in a book to
appear. The formula is derived from a bijection due to Bouttier, Di Francesco
and Guitter combined with a process (reminiscent of a construction of Pitman)
of aggregating connected components of a forest into a single tree. The formula
naturally extends to -constellations and quasi--constellations with
boundaries (the case corresponding to bipartite maps).Comment: 23 pages, full paper version of v1, with results extended to
constellations and quasi constellation
Local limits of uniform triangulations in high genus
We prove a conjecture of Benjamini and Curien stating that the local limits
of uniform random triangulations whose genus is proportional to the number of
faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in
arXiv:1401.3297. The proof relies on a combinatorial argument and the
Goulden--Jackson recurrence relation to obtain tightness, and probabilistic
arguments showing the uniqueness of the limit. As a consequence, we obtain
asymptotics up to subexponential factors on the number of triangulations when
both the size and the genus go to infinity.
As a part of our proof, we also obtain the following result of independent
interest: if a random triangulation of the plane is weakly Markovian in the
sense that the probability to observe a finite triangulation around the
root only depends on the perimeter and volume of , then is a mixture of
PSHT.Comment: 36 pages, 10 figure
On the mixing time of the flip walk on triangulations of the sphere
A simple way to sample a uniform triangulation of the sphere with a fixed
number of vertices is a Monte-Carlo method: we start from an arbitrary
triangulation and flip repeatedly a uniformly chosen edge. We give a lower
bound in on the mixing time of this Markov chain.Comment: 10 pages, 2 figures. Published versio
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