211 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
Counting self-avoiding walks
The connective constant of a graph is the asymptotic growth rate
of the number of self-avoiding walks on from a given starting vertex. We
survey three aspects of the dependence of the connective constant on the
underlying graph . Firstly, when is cubic, we study the effect on
of the Fisher transformation (that is, the replacement of vertices by
triangles). Secondly, we discuss upper and lower bounds for when
is regular. Thirdly, we present strict inequalities for the connective
constants of vertex-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
generator. Special prominence is given to open problems.Comment: Very minor changes for v2. arXiv admin note: text overlap with
arXiv:1301.309
Self-avoiding walks and amenability
The connective constant of an infinite transitive graph is the
exponential growth rate of the number of self-avoiding walks from a given
origin. The relationship between connective constants and amenability is
explored in the current work.
Various properties of connective constants depend on the existence of
so-called 'graph height functions', namely: (i) whether is a local
function on certain graphs derived from , (ii) the equality of and
the asymptotic growth rate of bridges, and (iii) whether there exists a
terminating algorithm for approximating to a given degree of accuracy.
In the context of amenable groups, it is proved that the Cayley graphs of
infinite, finitely generated, elementary amenable groups support graph height
functions, which are in addition harmonic. In contrast, the Cayley graph of the
Grigorchuk group, which is amenable but not elementary amenable, does not have
a graph height function.
In the context of non-amenable, transitive graphs, a lower bound is presented
for the connective constant in terms of the spectral bottom of the graph. This
is a strengthening of an earlier result of the same authors. Secondly, using a
percolation inequality of Benjamini, Nachmias, and Peres, it is explained that
the connective constant of a non-amenable, transitive graph with large girth is
close to that of a regular tree. Examples are given of non-amenable groups
without graph height functions, of which one is the Higman group.Comment: v2 differs from v1 in the inclusion of further material concerning
non-amenable graphs, notably an improved lower bound for the connective
constan
Extendable self-avoiding walks
The connective constant mu of a graph is the exponential growth rate of the
number of n-step self-avoiding walks starting at a given vertex. A
self-avoiding walk is said to be forward (respectively, backward) extendable if
it may be extended forwards (respectively, backwards) to a singly infinite
self-avoiding walk. It is called doubly extendable if it may be extended in
both directions simultaneously to a doubly infinite self-avoiding walk. We
prove that the connective constants for forward, backward, and doubly
extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB,
exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite,
strongly connected, quasi-transitive directed graph. The proofs rely on a 1967
result of Furstenberg on dimension, and involve two different arguments
depending on whether or not the graph is unimodular.Comment: Accepted versio
Locality of connective constants
The connective constant of a quasi-transitive graph is the
exponential growth rate of the number of self-avoiding walks from a given
origin. We prove a locality theorem for connective constants, namely, that the
connective constants of two graphs are close in value whenever the graphs agree
on a large ball around the origin (and a further condition is satisfied). The
proof exploits a generalized bridge decomposition of self-avoiding walks, which
is valid subject to the assumption that the underlying graph is
quasi-transitive and possesses a so-called unimodular graph height function
Cubic graphs and the golden mean
The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality for infinite, transitive, simple, cubic graphs, where is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for are presented for transitive cubic graphs with girth either or , and for certain quasi-transitive cubic graphs.This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1. ZL acknowledges support from the Simons Foundation under grant #351813 and the National Science Foundation under grant DMS-1608896
Self-avoiding walk on nonunimodular transitive graphs
We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length is comparable to the th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All these results apply in particular to the product of a -regular tree () with , for which these results were previously only known for large .Microsoft Researc
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