10,408 research outputs found
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
Percolation and isoperimetry on roughly transitive graphs
In this paper we study percolation on a roughly transitive graph G with
polynomial growth and isoperimetric dimension larger than one. For these graphs
we are able to prove that p_c < 1, or in other words, that there exists a
percolation phase. The main results of the article work for both dependent and
independent percolation processes, since they are based on a quite robust
renormalization technique. When G is transitive, the fact that p_c < 1 was
already known before. But even in that case our proof yields some new results
and it is entirely probabilistic, not involving the use of Gromov's theorem on
groups of polynomial growth. We finish the paper giving some examples of
dependent percolation for which our results apply.Comment: 32 pages, 2 figure
Graphs, permutations and topological groups
Various connections between the theory of permutation groups and the theory
of topological groups are described. These connections are applied in
permutation group theory and in the structure theory of topological groups.
The first draft of these notes was written for lectures at the conference
Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has
been corrected
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
Phase Transitions on Nonamenable Graphs
We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees
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