4,204 research outputs found
Coarse differentiation of quasi-isometries I: spaces not quasi-isometric to Cayley graphs
In this paper, we prove that certain spaces are not quasi-isometric to Cayley
graphs of finitely generated groups. In particular, we answer a question of
Woess and prove a conjecture of Diestel and Leader by showing that certain
homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely
generated group.
This paper is the first in a sequence of papers proving results announced in
[EFW0]. In particular, this paper contains many steps in the proofs of
quasi-isometric rigidity of lattices in Sol and of the quasi-isometry
classification of lamplighter groups. The proofs of those results are completed
in [EFW1].
The method used here is based on the idea of "coarse differentiation"
introduced in [EFW0].Comment: 44 pages; 4 figures; minor corrections addressing comments by the
refere
On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three
In the present paper we study forward Quantum Markov Chains (QMC) defined on
a Cayley tree. Using the tree structure of graphs, we give a construction of
quantum Markov chains on a Cayley tree. By means of such constructions we prove
the existence of a phase transition for the XY-model on a Cayley tree of order
three in QMC scheme. By the phase transition we mean the existence of two now
quasi equivalent QMC for the given family of interaction operators
.Comment: 34 pages, 1 figur
Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
We study the low energy asymptotics of periodic and random Laplace operators
on Cayley graphs of amenable, finitely generated groups. For the periodic
operator the asymptotics is characterised by the van Hove exponent or zeroth
Novikov-Shubin invariant. The random model we consider is given in terms of an
adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph.
The asymptotic behaviour of the spectral distribution is exponential,
characterised by the Lifshitz exponent. We show that for the adjacency
Laplacian the two invariants/exponents coincide. The result holds also for more
general symmetric transition operators. For combinatorial Laplacians one has a
different universal behaviour of the low energy asymptotics of the spectral
distribution function, which can be actually established on quasi-transitive
graphs without an amenability assumption. The latter result holds also for long
range bond percolation models
Sensitivity of mixing times of Cayley graphs
We show that the total variation mixing time is neither quasi-isometry
invariant nor robust, even for Cayley graphs. The Cayley graphs serving as an
example have unbounded degrees. For non-transitive graphs we show bounded
degree graphs for which the mixing time from the worst point for one graph is
asymptotically smaller than the mixing time from the best point of the random
walk on a network obtained by increasing some of the edge weights from 1 to
.Comment: 28 pages, 1 figur
Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs
Central limit theorems for linear statistics of lattice random fields
(including spin models) are usually proven under suitable mixing conditions or
quasi-associativity. Many interesting examples of spin models do not satisfy
mixing conditions, and on the other hand, it does not seem easy to show central
limit theorem for local statistics via quasi-associativity. In this work, we
prove general central limit theorems for local statistics and exponentially
quasi-local statistics of spin models on discrete Cayley graphs with polynomial
growth. Further, we supplement these results by proving similar central limit
theorems for random fields on discrete Cayley graphs and taking values in a
countable space but under the stronger assumptions of {\alpha}-mixing (for
local statistics) and exponential {\alpha}-mixing (for exponentially
quasi-local statistics). All our central limit theorems assume a suitable
variance lower bound like many others in the literature. We illustrate our
general central limit theorem with specific examples of lattice spin models and
statistics arising in computational topology, statistical physics and random
networks. Examples of clustering spin models include quasi-associated spin
models with fast decaying covariances like the off-critical Ising model, level
sets of Gaussian random fields with fast decaying covariances like the massive
Gaussian free field and determinantal point processes with fast decaying
kernels. Examples of local statistics include intrinsic volumes, face counts,
component counts of random cubical complexes while exponentially quasi-local
statistics include nearest neighbour distances in spin models and Betti numbers
of sub-critical random cubical complexes.Comment: Minor changes incorporated based on suggestions by referee
Geometry of generated groups with metrics induced by their Cayley color graphs
Let be a group and let be a generating set of . In this article,
we introduce a metric on with respect to , called the cardinal
metric. We then compare geometric structures of and ,
where denotes the word metric. In particular, we prove that if is
finite, then and are not quasi-isometric in the case when
has infinite diameter and they are bi-Lipschitz equivalent
otherwise. We also give an alternative description of cardinal metrics by using
Cayley color graphs. It turns out that color-permuting and color-preserving
automorphisms of Cayley digraphs are isometries with respect to cardinal
metrics
Geodesically Tracking Quasi-geodesic Paths for Coxeter Groups
The main theorem of this paper classifies the quasi-geodesics in a Coxeter
group that are tracked by geodesics. As corollaries, we show that if a Coxeter
group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are
tracked by Cayley graph geodesics, all special subgroups of the Coxeter group
are quasi-convex in X, and in Cayley graphs for Coxeter groups, elements of
infinite order are tracked by geodesics.Comment: 15 pages, 5 figure
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
- …