37,368 research outputs found
On the Genus of Random Regular Graphs
The orientable genus of a graph is the minimum number of handles required
to embed that graph on a surface. Determining graph genus is a fundamental yet
often difficult task. We show that, for any integer , the genus of a
random -regular graph on nodes is
with high probability for any Comment: 3 page
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
Fix an ordinary abelian variety defined over a finite field. The ideal class
group of its endomorphism ring acts freely on the set of isogenous varieties
with same endomorphism ring, by complex multiplication. Any subgroup of the
class group, and generating set thereof, induces an isogeny graph on the orbit
of the variety for this subgroup. We compute (under the Generalized Riemann
Hypothesis) some bounds on the norms of prime ideals generating it, such that
the associated graph has good expansion properties.
We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and
Robert for computing explicit isogenies in genus 2, to prove random
self-reducibility of the discrete logarithm problem within the subclasses of
principally polarizable ordinary abelian surfaces with fixed endomorphism ring.
In addition, we remove the heuristics in the complexity analysis of an
algorithm of Galbraith for explicitly computing isogenies between two elliptic
curves in the same isogeny class, and extend it to a more general setting
including genus 2.Comment: 18 page
Directed Random Walk on the Lattices of Genus Two
The object of the present investigation is an ensemble of self-avoiding and
directed graphs belonging to eight-branching Cayley tree (Bethe lattice)
generated by the Fucsian group of a Riemann surface of genus two and embedded
in the Pincar\'e unit disk. We consider two-parametric lattices and calculate
the multifractal scaling exponents for the moments of the graph lengths
distribution as functions of these parameters. We show the results of numerical
and statistical computations, where the latter are based on a random walk
model.Comment: 17 pages, 8 figure
Scaling Limits for Random Quadrangulations of Positive Genus
We discuss scaling limits of large bipartite quadrangulations of positive
genus. For a given , we consider, for every , a random
quadrangulation \q_n uniformly distributed over the set of all rooted
bipartite quadrangulations of genus with faces. We view it as a metric
space by endowing its set of vertices with the graph distance. We show that, as
tends to infinity, this metric space, with distances rescaled by the factor
, converges in distribution, at least along some subsequence, toward
a limiting random metric space. This convergence holds in the sense of the
Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of
the choice of the subsequence, the Hausdorff dimension of the limiting space is
almost surely equal to 4. Our main tool is a bijection introduced by Chapuy,
Marcus, and Schaeffer between the quadrangulations we consider and objects they
call well-labeled -trees. An important part of our study consists in
determining the scaling limits of the latter
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Height representation of XOR-Ising loops via bipartite dimers
The XOR-Ising model on a graph consists of random spin configurations on
vertices of the graph obtained by taking the product at each vertex of the
spins of two independent Ising models. In this paper, we explicitly relate loop
configurations of the XOR-Ising model and those of a dimer model living on a
decorated, bipartite version of the Ising graph. This result is proved for
graphs embedded in compact surfaces of genus g.
Using this fact, we then prove that XOR-Ising loops have the same law as
level lines of the height function of this bipartite dimer model. At
criticality, the height function is known to converge weakly in distribution to
a Gaussian free field.
As a consequence, results of this paper shed a light on the occurrence of the
Gaussian free field in the XOR-Ising model. In particular, they prove a
discrete analogue of Wilson's conjecture, stating that the scaling limit of
XOR-Ising loops are "contour lines" of the Gaussian free field.Comment: 41 pages, 10 figure
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