1,294 research outputs found
Constructive field theory without tears
We propose to treat the Euclidean theory constructively in a simpler
way. Our method, based on a new kind of "loop vertex expansion", no longer
requires the painful intermediate tool of cluster and Mayer expansions.Comment: 22 pages, 10 figure
Counting the spanning trees of the 3-cube using edge slides
We give a direct combinatorial proof of the known fact that the 3-cube has
384 spanning trees, using an "edge slide" operation on spanning trees. This
gives an answer in the case n=3 to a question implicitly raised by Stanley. Our
argument also gives a bijective proof of the n=3 case of a weighted count of
the spanning trees of the n-cube due to Martin and Reiner.Comment: 17 pages, 9 figures. v2: Final version as published in the
Australasian Journal of Combinatorics. Section 5 shortened and restructured;
references added; one figure added; some typos corrected; additional minor
changes in response to the referees' comment
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
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
The simplicial boundary of a CAT(0) cube complex
For a CAT(0) cube complex , we define a simplicial flag complex
, called the \emph{simplicial boundary}, which is a
natural setting for studying non-hyperbolic behavior of . We compare
to the Roller, visual, and Tits boundaries of
and give conditions under which the natural CAT(1) metric on
makes it (quasi)isometric to the Tits boundary.
allows us to interpolate between studying geodesic
rays in and the geometry of its \emph{contact graph} , which is known to be quasi-isometric to a tree, and we characterize
essential cube complexes for which the contact graph is bounded. Using related
techniques, we study divergence of combinatorial geodesics in using
. Finally, we rephrase the rank-rigidity theorem of
Caprace-Sageev in terms of group actions on and
and state characterizations of cubulated groups with
linear divergence in terms of and .Comment: Lemma 3.18 was not stated correctly. This is fixed, and a minor
adjustment to the beginning of the proof of Theorem 3.19 has been made as a
result. Statements other than 3.18 do not need to change. I thank Abdul
Zalloum for the correction. See also: arXiv:2004.01182 (this version differs
from previous only by addition of the preceding link, at administrators'
request
Convergence of three-dimensional loop-erased random walk in the natural parametrization
In this work, we consider loop-erased random walk (LERW) and its scaling
limit in three dimensions, and prove that 3D LERW parametrized by renormalized
length converges to its scaling limit parametrized by some suitable measure
with respect to the uniform convergence topology in the lattice size scaling
limit. Our result improves the previous work of Gady Kozma (Acta Math.
199(1):29-152), which shows that the rescaled trace of 3D LERW converges weakly
to a random compact set with respect to the Hausdorff distance.Comment: 74 pages, 3 figure
- …