9 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
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
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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Connective constants and height functions for Cayley graphs
The connective constant () of an infinite transitive graph is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved 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. A condition of the theorem was that the graphs support so-called “unimodular graph height functions”. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a “ height function”. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency.
It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic.
Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.The first author was supported in part by EPSRC Grant EP/I03372X/1. The second author was supported in part by Simons Collaboration Grant #351813 and NSF grant #1608896
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 is ballistic on graphs with more than one end
We prove that on any transitive graph with infinitely many ends, a self-avoiding walk of length is ballistic with extremely high probability, in the sense that there exist constants c,t>0 such that for every . Furthermore, we show that the number of self-avoiding walks of length grows asymptotically like , in the sense that there exists C>0 such that for every . Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of which does not fix an end of