6,297 research outputs found
Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time
For critical bond-percolation on high-dimensional torus, this paper proves
sharp lower bounds on the size of the largest cluster, removing a logarithmic
correction in the lower bound in Heydenreich and van der Hofstad (2007). This
improvement finally settles a conjecture by Aizenman (1997) about the role of
boundary conditions in critical high-dimensional percolation, and it is a key
step in deriving further properties of critical percolation on the torus.
Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on
diameter and mixing time of the largest clusters. We further prove that the
volume bounds apply also to any finite number of the largest clusters. The main
conclusion of the paper is that the behavior of critical percolation on the
high-dimensional torus is the same as for critical Erdos-Renyi random graphs.
In this updated version we incorporate an erratum to be published in a
forthcoming issue of Probab. Theory Relat. Fields. This results in a
modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming
issue of Probab. Theory Relat. Field
Metastability for the contact process on the preferential attachment graph
We consider the contact process on the preferential attachment graph. The
work of Berger, Borgs, Chayes and Saberi [BBCS1] confirmed physicists
predictions that the contact process starting from a typical vertex becomes
endemic for an arbitrarily small infection rate with positive
probability. More precisely, they showed that with probability , it survives for a time exponential in the largest degree. Here we obtain
sharp bounds for the density of infected sites at a time close to exponential
in the number of vertices (up to some logarithmic factor).Comment: 45 pages; accepted for publication in Internet Mathematic
On two conjectures about the proper connection number of graphs
A path in an edge-colored graph is called proper if no two consecutive edges
of the path receive the same color. For a connected graph , the proper
connection number of is defined as the minimum number of colors
needed to color its edges so that every pair of distinct vertices of are
connected by at least one proper path in . In this paper, we consider two
conjectures on the proper connection number of graphs. The first conjecture
states that if is a noncomplete graph with connectivity and
minimum degree , then , posed by Borozan et al.~in
[Discrete Math. 312(2012), 2550-2560]. We give a family of counterexamples to
disprove this conjecture. However, from a result of Thomassen it follows that
3-edge-connected noncomplete graphs have proper connection number 2. Using this
result, we can prove that if is a 2-connected noncomplete graph with
, then , which solves the second conjecture we want to
mention, posed by Li and Magnant in [Theory \& Appl. Graphs 0(1)(2015), Art.2].Comment: 10 pages. arXiv admin note: text overlap with arXiv:1601.0416
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture
We introduce a new combinatorial abstraction for the graphs of polyhedra. The
new abstraction is a flexible framework defined by combinatorial properties,
with each collection of properties taken providing a variant for studying the
diameters of polyhedral graphs. One particular variant has a diameter which
satisfies the best known upper bound on the diameters of polyhedra. Another
variant has superlinear asymptotic diameter, and together with some
combinatorial operations, gives a concrete approach for disproving the Linear
Hirsch Conjecture.Comment: 16 pages, 4 figure
Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction
For random collections of self-avoiding loops in two-dimensional domains, we
define a simple and natural conformal restriction property that is
conjecturally satisfied by the scaling limits of interfaces in models from
statistical physics. This property is basically the combination of conformal
invariance and the locality of the interaction in the model. Unlike the Markov
property that Schramm used to characterize SLE curves (which involves
conditioning on partially generated interfaces up to arbitrary stopping times),
this property only involves conditioning on entire loops and thus appears at
first glance to be weaker.
Our first main result is that there exists exactly a one-dimensional family
of random loop collections with this property---one for each k in (8/3,4]---and
that the loops are forms of SLE(k). The proof proceeds in two steps. First,
uniqueness is established by showing that every such loop ensemble can be
generated by an "exploration" process based on SLE.
Second, existence is obtained using the two-dimensional Brownian loop-soup,
which is a Poissonian random collection of loops in a planar domain. When the
intensity parameter c of the loop-soup is less than 1, we show that the outer
boundaries of the loop clusters are disjoint simple loops (when c>1 there is
almost surely only one cluster) that satisfy the conformal restriction axioms.
We prove various results about loop-soups, cluster sizes, and the c=1 phase
transition.
Taken together, our results imply that the following families are equivalent:
1. The random loop ensembles traced by certain branching SLE(k) curves for k
in (8/3, 4].
2. The outer-cluster-boundary ensembles of Brownian loop-soups for c in (0,
1].
3. The (only) random loop ensembles satisfying the conformal restriction
axioms.Comment: This 91 page-long paper contains the previous versions (v2) of both
papers arxiv:1006.2373 and arxiv:1006.2374 that correspond to Part I and Part
II of the present paper. This merged longer paper is to appear in Annals of
Mathematic
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