35 research outputs found
On balanced planar graphs, following W. Thurston
Let be an orientation-preserving branched covering map of
degree , and let be an oriented Jordan curve passing through
the critical values of . Then is an oriented graph
on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced
balanced planar graphs and showed that they combinatorially characterize all
such , where has distinct critical values. We give a
detailed account of this discussion, along with some examples and an appendix
about Hurwitz numbers.Comment: 17 page
Random surfaces and Liouville quantum gravity
Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces which can be thought of as the canonical models of random two-dimensional Riemannian manifolds, in the same sense that Brownian motion is the canonical model of a random path. LQG surfaces are the continuum limits of discrete random surfaces called random planar maps.
In this expository article, we discuss the definition of random planar maps and LQG, the sense in which random planar maps converge to LQG, and the motivations for studying these objects. We also mention several open problems.
We do not assume any background knowledge beyond that of a second-year mathematics graduate student
Basic properties of the infinite critical-FK random map
We investigate the critical Fortuin-Kasteleyn (cFK) random map model. For
each and integer , this model chooses a planar map
of edges with a probability proportional to the partition function of
critical -Potts model on that map. Sheffield introduced the
hamburger-cheeseburger bijection which maps the cFK random maps to a family of
random words, and remarked that one can construct infinite cFK random maps
using this bijection. We make this idea precise by a detailed proof of the
local convergence. When , this provides an alternative construction of the
UIPQ. In addition, we show that the limit is almost surely one-ended and
recurrent for the simple random walk for any , and mutually singular in
distribution for different values of .Comment: 14 pages, 6 figures. v2: Fixed the proof of main theorem, removed old
lemma 5, added results on mutually singular measures and ergodicity.
Submitted to Annales de l'Institut Henri Poincar\'e
Orienting triangulations
We prove that any triangulation of a surface different from the sphere and
the projective plane admits an orientation without sinks such that every vertex
has outdegree divisible by three. This confirms a conjecture of Bar\'at and
Thomassen and is a step towards a generalization of Schnyder woods to higher
genus surfaces
A simple model of trees for unicellular maps
We consider unicellular maps, or polygon gluings, of fixed genus. A few years
ago the first author gave a recursive bijection transforming unicellular maps
into trees, explaining the presence of Catalan numbers in counting formulas for
these objects. In this paper, we give another bijection that explicitly
describes the "recursive part" of the first bijection. As a result we obtain a
very simple description of unicellular maps as pairs made by a plane tree and a
permutation-like structure. All the previously known formulas follow as an
immediate corollary or easy exercise, thus giving a bijective proof for each of
them, in a unified way. For some of these formulas, this is the first bijective
proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and
the Goupil-Schaeffer formula. We also discuss several applications of our
construction: we obtain a new proof of an identity related to covered maps due
to Bernardi and the first author, and thanks to previous work of the second
author, we give a new expression for Stanley character polynomials, which
evaluate irreducible characters of the symmetric group. Finally, we show that
our techniques apply partially to unicellular 3-constellations and to related
objects that we call quasi-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a
refinement by degree of the Harer-Zagier formula and more details in some
proof