30,223 research outputs found
Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions
A general formulation is presented for continuum scaling limits of stochastic
spanning trees. A spanning tree is expressed in this limit through a consistent
collection of subtrees, which includes a tree for every finite set of endpoints
in . Tightness of the distribution, as , is established for
the following two-dimensional examples: the uniformly random spanning tree on
, the minimal spanning tree on (with random edge
lengths), and the Euclidean minimal spanning tree on a Poisson process of
points in with density . In each case, sample trees are
proven to have the following properties, with probability one with respect to
any of the limiting measures: i) there is a single route to infinity (as was
known for ), ii) the tree branches are given by curves which are
regular in the sense of H\"older continuity, iii) the branches are also rough,
in the sense that their Hausdorff dimension exceeds one, iv) there is a random
dense subset of , of dimension strictly between one and two, on the
complement of which (and only there) the spanning subtrees are unique with
continuous dependence on the endpoints, v) branching occurs at countably many
points in , and vi) the branching numbers are uniformly bounded. The
results include tightness for the loop erased random walk (LERW) in two
dimensions. The proofs proceed through the derivation of scale-invariant power
bounds on the probabilities of repeated crossings of annuli.Comment: Revised; 54 pages, 6 figures (LaTex
How to Couple from the Past Using a Read-Once Source of Randomness
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure
An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles
The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It
gives a sufficient condition on a probability space and a collection of events
for the existence of an outcome that simultaneously avoids all of those events.
Finding such an outcome by an efficient algorithm has been an active research
topic for decades. Breakthrough work of Moser and Tardos (2009) presented an
efficient algorithm for a general setting primarily characterized by a product
structure on the probability space.
In this work we present an efficient algorithm for a much more general
setting. Our main assumption is that there exist certain functions, called
resampling oracles, that can be invoked to address the undesired occurrence of
the events. We show that, in all scenarios to which the original Lovasz Local
Lemma applies, there exist resampling oracles, although they are not
necessarily efficient. Nevertheless, for essentially all known applications of
the Lovasz Local Lemma and its generalizations, we have designed efficient
resampling oracles. As applications of these techniques, we present new results
for packings of Latin transversals, rainbow matchings and rainbow spanning
trees.Comment: 47 page
Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes
A well-known conjecture of Richard Stanley posits that the -vector of the
independence complex of a matroid is a pure -sequence. The
conjecture has been established for various classes but is open for graphic
matroids. A biconed graph is a graph with two specified `coning vertices', such
that every vertex of the graph is connected to at least one coning vertex. The
class of biconed graphs includes coned graphs, Ferrers graphs, and complete
multipartite graphs. We study the -vectors of graphic matroids arising from
biconed graphs, providing a combinatorial interpretation of their entries in
terms of `edge-rooted forests' of the underlying graph. This generalizes
constructions of Kook and Lee who studied the M\"obius coinvariant (the last
nonzero entry of the -vector) of graphic matroids of complete bipartite
graphs. We show that allowing for partially edge-rooted forests gives rise to a
pure multicomplex whose face count recovers the -vector, establishing
Stanley's conjecture for this class of matroids.Comment: 15 pages, 3 figures; V2: added omitted author to metadat
Robustness: a New Form of Heredity Motivated by Dynamic Networks
We investigate a special case of hereditary property in graphs, referred to
as {\em robustness}. A property (or structure) is called robust in a graph
if it is inherited by all the connected spanning subgraphs of . We motivate
this definition using two different settings of dynamic networks. The first
corresponds to networks of low dynamicity, where some links may be permanently
removed so long as the network remains connected. The second corresponds to
highly-dynamic networks, where communication links appear and disappear
arbitrarily often, subject only to the requirement that the entities are
temporally connected in a recurrent fashion ({\it i.e.} they can always reach
each other through temporal paths). Each context induces a different
interpretation of the notion of robustness.
We start by motivating the definition and discussing the two interpretations,
after what we consider the notion independently from its interpretation, taking
as our focus the robustness of {\em maximal independent sets} (MIS). A graph
may or may not admit a robust MIS. We characterize the set of graphs \forallMIS
in which {\em all} MISs are robust. Then, we turn our attention to the graphs
that {\em admit} a robust MIS (\existsMIS). This class has a more complex
structure; we give a partial characterization in terms of elementary graph
properties, then a complete characterization by means of a (polynomial time)
decision algorithm that accepts if and only if a robust MIS exists. This
algorithm can be adapted to construct such a solution if one exists
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